It's possible, but not with the resources you listed above.
Mirror move is a WotC web supplement, likely turned into the spell compendium's Heroics. Both feats will provide a leavening of "free" feats for your character. Enchanting them into a magic item using the custom magic item rules is technically possible, though may get dice thrown at you.
You may also find this thread useful:
If so, get Magical Training (PGtF) to meet the arcane casting prerequisite of Dweomerkeeper, ... Also be sure to pick up Initiate of Mystra (PGtF) at some point if possible, though it's typically delayed until around 9th level.
If you can use flaws, use one of the elf races, and later on you can (Su) Dark Chaos Shuffle your four racial marital weapon proficiency feats into four Extra Turning feats to power DMM: Persist. If you're doing this, you should also spend some starting gold on one or more bonus feats from visiting the special locations detailed in Complete Scoundrel, which can also be Dark Chaos Shuffled to something more useful.
Unfortunately, all of these suggestions rely on books outside your "allowed book" list. It also seems the class has been changed in the web edition versus the book edition, so your mileage may vary. Also, given that the thought bottle is in Complete Arcane, don't worry too much about supernatural spell. As the class doesn't grant dual casting, I would try to find a feat-replacement for that level of wizard.
Still, with the set above, I expect much of what you'll find is a function of negotiation with your DM. Having mirror move as a mantled spell is not a horrible idea and, as it costs you time and slots to cast, isn't absurdly broken (just mostly.)
Caster Level Looping
I'm going to open this answer by saying - there is no definitive rules text on how to treat, or in what order, stacking caster levels from classes or prestige classes. There is a general rule from the 3.5e FAQ that states that 'in general' one should apply bonuses in order most beneficial to the creature, but it's not exactly specific, or definitive, especially in this edge case.
Also, Caster Level Looping And You (by Bauglir) has how the CO people think all this works, also, nearly every way to multiplicatively increase Caster Level in the game.
It's worth noting that there are two ways to read this. One (the one CO likes) is that 'spellcaster level' is referring to 'caster level', and indeed, this is actually the way the game parses it in other abilities. But it could also be referring to 'levels in spellcasting classes', which means basically NONE of the shenanigans outlined here will do anything.
In my answer, I will be assuming the former, due to the way the question is asked. In the case of the latter, simply count up your class levels in arcane casting classes (which doesn't include PrC's that say '+1 level in existing blah', tough luck), and that's your CL for everything.
It's worth noting that former reading is not suitable for play or practical optimization in any sense. Neither are any of the other caster level loops, even (or especially) the ones that are written more explicitly.
But on to the questions.
Question 1
Alex is a Spellthief 2 / Wizard 16 / Green Star Adept 2 who possesses the feat Master Spellthief. What's Alex's caster level for his wizard spells?
Alex's Master Spellthief feat means his Spellthief CL is 18 (16 Wizard + 2 Spellthief) and his Wizard CL is 18 (''), before he takes any levels of Green Star Adept, 'when determining caster level for arcane spells'. So for arcane spells it's 18, otherwise it's the normal numbers - 2 for spellthief, 16 for wizard. This is fine, as the numeric effects of Arcane Spells is kind of what you want to use a caster level for as an arcane caster. But it's important to note that his caster level is still just 2, and 16, not 18.
So when he takes Green Star Adept, he adds his Green Star Adept levels (2) to his Wizard levels (16) to get his new Wizard 'effective caster level' (18). So he now has three things related to being a Wizard;
His Actual Wizard Caster Level: 16
His 'Effective' Wizard Caster Level: 18
His Wizard Caster Level For the Purposes of Casting Arcane Spells: 20.
There's no reason why 'effective caster level' would not be counted as a caster level for the purposes of Master Spellthief, so 2 + 18 = 20 = what you use as your CL for arcane spells cast either from the Spellthief class or the Wizard class.
Additionally, the Wizard has the spells known and per day of a 17th level Wizard, due to Green Star Adept's Spells per Day/Spells Known class feature.
Question 2
Bob is a Spellthief 4 / Wizard 15 / Knight of the Weave 1 who possesses the feat Master Spellthief. What's Bob's caster level for his A) spellthief spells, B) wizard spells, and C) knight of the weave spells?
And here starteth le fromage. Basically, Knight of the Weave's caster level is equal to Bob's Spellthief AND Wizard caster levels added together, but again NOT the Master Spellthief total, which is purely for the purposes of casting arcane spells only. So 4 + 15 = Knight of the Weave Caster Level of 19.
But whenever Bob casts an arcane spell, Master Spellthief kicks in. And it goes 'hmm what caster levels you got', finds Spellthief at 4, Wizard at 15, and some wacky prestige class called Knight of the Weave at 19, and then adds all those numbers together because they are all 'arcane spellcaster levels' and comes out with the absurd number of caster level of 38 for the purposes of casting arcane spells.
Question 3
Chris is a Bard 1 / Spellthief 1 / Wizard 8 / Sublime Chord 10 who possesses the feat Master Spellthief. She applies the sublime chord class feature spells per day to her wizard casting. What's Chris's caster level for A) wizard spells and B) sublime chord spells?
Sublime Chord sets all arcane casting classes Caster Level to 18, in this scenario, due to the wording of it's 'spells per day' ability.
Then along comes our good friend Master Spellthief, sees 4 Caster Levels (Bard, Spellthief, Wizard, Sublime Chord) set at 18. It goes, you know what? Let's add all those together, and end up with a CL for the purposes of casting Arcane Spells of 72.
Then everyone cries.
Question 4
Dylan is a Sorcerer 1 / Spellthief 4 / Wizard 3 / Ultimate Magus 10 who possesses the feat Master Spellthief. He applies the ultimate magus class feature spellcasting and extraordinary ability arcane spell power to his wizard casting when he can and his sorcerer casting when he must. What's Dylan's caster level for his A) wizard spells and B) sorcerer spells?
He starts out with Sorc CL 10, Spellthief CL 4, Wizard CL 11. He possesses a +4 to CL for the purposes of arcane spells. Master Spellthief turns his CL for all his arcane casting classes for the purposes of casting arcane spells into 25 (4+11+10). To which you then add +4. Master Spellthief (under the nice reading) sums the totals of your caster levels, not the totals of your 'caster level for the purposes of arcane spells', so UM's ability simply adds, it doesn't get multiplied across.
Question 5
Ethan is a Bard 1 / Sorcerer 1 / Spellthief 2/ Wizard 3 / Green Star Adept 2 / Knight of the Weave 1 / Sublime Chord 6 / Ultimate Magus 4 (or another more favorable combination you choose) with the feat Master Spellthief. He applies everything he can to his wizard casting until he can apply it to his sublime chord casting and everything he must to his wizard casting or his sorcerer casting, favoring wizard. What are his caster levels for his A) bard spells, B) sorcerer spells, C) wizard spells, D) knight of the weave spells, and E) sublime chord spells?
Why are you doing this to me.
Right. Following the FAQ guideline about adding things in most beneficial order, this is actually relatively simple. Ultimate Magus advances Sublime Chord 2 steps (you'd be better off just taking more wizard levels, you only care about highest non-sublime-chord CL when sublime chord is in the mix, and dipping as many other arcane casting classes as possible), putting total 'sublime chord levels' at 8. Sublime Chord adds itself to Wizard to determine your CL for ALL arcane spellcasting clsases, so all your arcane spellcasting classes now have a CL of 3+8=11. Green Star Adept adds +2 to Sublime Chord on top, so you end up with a CL of 13 on all your arcane casting classes.
Knight of the Weave now does it's thing. You can't have it do it's thing 'first' and then add sublime chord to it and then have everything be CL 22 or something because Sublime Chord adds itself to 'level in a spellcasting class', not 'CL'. So '+1 existing level' prestige classes etc mean nothing to Sublime Chord unless they are advancing Sublime Chord, since it determines it's CL based on class levels in another class + Sublime Chord levels/prestige class bonuses/stuff. Anyway, you've got 5 caster levels (Bard, Sorcerer, Wizard, Spellthief, Sublime Chord) and Sublime has set them all to 13. So, 5x13=65, +1 for your Knight of the Weave level, gives a Knight of the Weave CL of 66.
Master Spellthief then does it's thing when you go to cast a spell, as normal. 66+65= CL 131.
Then Ultimate Magus does it's Arcane Power thing for +2 more, giving you a final total of
CL 133.
Important Concepts for Understanding This Stuff
Prestige classes that advance spellcasting do not themselves have a 'caster level' - they simply modify the caster level and spells known etc of another class.
'Caster level' is not the same as 'caster level for the purposes of arcane spells', it's like 'hp while raging' or 'hp while buffed with Aid' is not the same as 'base hp'.
Sublime Chord finds a base class, adds that base class' class levels to Sublime Chord levels, adds any randumb CL modifiers (Orange Ioun Stone, Green Star Adept) then sets every arcane CL the character has to that number, including it's own. It's not adding or involving CLs at all, just class level + class level + magic items or spells or whatever. Prestige classes that say '+1 spellcaster level' that apply to sublime chord count, but not those that apply to the class it is adding to itself.
Knight of the Weave doesn't do any of that, it just finds all the CLs around, adds them together, and uses that as it's own CL.
Master Spellthief doesn't ever modify any CLs. It just adds all existing CLs together to get a number used AS the CL for casting arcane spells. It doesn't actually change the actual CLs the classes have, just the number you use to cast spells.
'Effective caster level' is mostly the same as 'caster level' since you use it for anything involving caster level.
You do all of these transformations only once, in whatever order you like, but you can't just have them feeding back into each other forever due to the way they're written.
If one of these guys with Sublime Chord or Knight of the Weave picked up an Orange Ioun Stone, it would increment all their numbers and therefore change Sublime Chord's CL or Knight of the Weave's CL. You'd go back to the start and apply all the bonuses again, but this time including the Orange Ioun Stone, in whatever order was most advantageous to you as normal. So you'd undo the transformations and go from the top, work it all out again but with the new factor. Same if you gain a level, etc.
Best Answer
One, your stats are basically incapable of supporting any character who isn’t a pure spellcaster. For a pure spellcaster, those are pretty good stats, but for anyone else they’re painfully poor. An 8 in Constitution is putting your life in grave danger all the time. This is one of many reasons why I recommend against rolled ability scores. Have you spoken to your DM about this problem? Maybe you can convince him to allow a reroll or moving to an ability array or something.
Alternatively, you could maybe retrain as a spellcaster, but that’s not going to mesh well with your interest in archery. Bizarrely, 3.5 has very poor support for archery; basically all of the archery-related prestige classes are close to worthless, and archery requires a ton of really weak feats to get anything done (Point-Blank Shot is garbage but required for everything, Precise Shot is necessary but all it does is eliminate a painful penalty, Manyshot is incompatible with Rapid Shot, which is finally half-decent except you’ve burned four feats on a half-decent benefit!).
Finally, it’s conceivable that you might have more hope if your DM expanded that extremely-narrow field of available books. As is, all the best options I can think of for making use of that Dexterity (e.g. Crossbow Sniper, Dead Eye, Targeteer) are unavailable to you (examples are from Player’s Handbook II, Dragon Compendium, and Dragon vol. 310, respectively). And banning psionics means some very solid Wisdom-based magical-warrior options (ardent, psychic warrior, war mind) are unavailable to you. At the very least, Spell Compendium would mean that you actually have decent spells; as it is, you are literally weaker than a druid of half your level.
Anyway, taking what you’ve got as set in stone...
Sword of the Arcane Order is worthless to you because your Intelligence is a mere 10. It’s a great feat, but your stats need to be different.
This becomes a different story if your Intelligence goes up; then it’s the best feat you can take. Must-have if your DM allows you to change your stats, for instance. Assuming you are the only elf in the party, getting a special elf-only headband of intelligence as a reward for something might also do the trick, if the bonus is large enough.
Alternatively, you might ask your DM for a Wisdom-based analogue to Sword of the Arcane Order. Under normal circumstances, the requirement for Intelligence is something of a balancing factor to SotAO, but you really could use all the help you can get.
Sword of the Arcane Order does very awkwardly just... stop at 11th level. Low Intelligence, particularly if all your Intelligence comes from an item (i.e. you can’t improve it through another item) exacerbates this. That’s probably fine as most campaigns don’t go that high, and frankly I’m not sure there’s anything you can do to maintain a decent power level much past there anyway.
Order of the Bow Initiate is a complete waste of time.
Depending on how many levels you’ve burned on ranger, multiclassing to cleric or druid might be an option. Your Wisdom is very low for such classes, but even so their spells offer far more than ranger ever will.
Cleric would allow you to take Divine feats; you’ll miss Complete Champion quite a bit, but Complete Divine still has some good ones. I would strongly consider going for the Planning and Undeath domains, even though those are probably difficult to justify for your character. If you can think of any reason your character might get those, though, they would help you: you would get Extra Turning and Extend Spell, and could take Persistent Spell and Divine Metamagic (Complete Divine) in it so at least one spell a day is active all day. Divine power is a common choice but not the only option.
If you cannot justify the Undeath domain (likely), consider taking the Elf domain and asking if, since both the domain and ranger are giving you the same bonus feat, you could get Extra Turning as a bonus feat. Giving alternate bonus feats to replace duplicate bonus feats is pretty common.
A cloak of charisma will help a bit. You might even consider bumping your Charisma to 10 with your 4th-level or 8th-level ability score bump, but only do that if you don’t think you’ll reach the next bump; Wisdom is still the higher priority. In other words, Wis 14/Cha 10 is better than Wis 15/Cha 9, but Wis 16/Cha 9 is way better than Wis 15/Cha 10.
Druid would be awkward, considering that most Wild Shape forms are going to nix your archery. Sticking to primate forms might work, however; you could talk to your DM. But even allowing that, Wild Shape replaces your physical ability scores, completely wasting your Dexterity. You’d still be stronger than you are, but it would be awkward. The advantage is that it’s much less of a leap than cleric, since ranger is basically half a druid.
The divine crusader prestige class in Complete Divine also might be a solid option. It has full Base Attack Bonus, so it works out well enough for your archery, and it gets 9th-level spells in 10 levels. That’s great. The drawback is, the divine crusader’s spell list consists of exactly one domain. Choose wisely.
Unfortunately, without Spell Compendium you are quite limited here. War domain may end up being your best bet, and that’s kind of sad. Travel is another decent-ish option. You’ll also have to beg your DM to let you use Wisdom instead of Charisma.
The other class features of the divine crusader are pretty much garbage though; OK protection from electricity and acid damage, but that’s about it. It does get Weapon Specialization in its deity’s favored weapon, and requires Weapon Focus in the same, so hopefully you worship a deity that uses some kind of bow. Ask your DM if you can waive the Weapon Focus requirement if you choose the War domain; the domain grants that feat anyway, so it would be a complete waste to you.
A single level of cleric before divine crusader is also a decent idea. You could use the War domain to get into divine crusader in the first place, and Turn Undead remains very useful for Divine feats.
Sacred exorcist, also Complete Divine, can get you Turn Undead without having to dip cleric. Requires you be able to cast dismissal or dispel evil, though, which would put a harsh requirement on your domain choice.
At higher levels, dipping a single level of contemplative and/or divine oracle (both Complete Divine) can allow you to add more domains to divine crusader. That dramatically increases your spell list. Divine oracle 2 also has a very nice version of Evasion that works in all armor.
Unfortunately, those are all the options I can offer to you. If you do get additional supplements made available, Complete Champion, Player’s Handbook II, Spell Compendium, and Tome of Battle are probably your best bets. Expanded Psionics Handbook and Complete Psionic would also help. Unfortunately, it’s quite unlikely that you’ll get Dragon material, which is a shame, since it’d help.