First question: trigger

The character needs only to have at least one level of a class which can cast the spell. In this case a first level wizard will be able to trigger any wizard spell.

Second question: completion

From the SRD: Scrolls

To have any chance of activating a scroll spell, the scroll user must meet the following requirements.

• The spell must be of the correct type (arcane or divine). Arcane spellcasters (wizards, sorcerers, and bards) can only use scrolls containing arcane spells, and divine spellcasters (clerics, druids, paladins, and rangers) can only use scrolls containing divine spells. (The type of scroll a character creates is also determined by his class.)
• The user must have the spell on her class list.
• The user must have the requisite ability score. If the user meets all the requirements noted above, and her caster level is at least equal to the spell's caster level, she can automatically activate the spell without a check.

This means that the character must be a spell caster of the correct type (arcane or divine): so no a fighter will not be able to use a scroll.

If she meets all three requirements but her own caster level is lower than the scroll spell's caster level, then she has to make a caster level check (DC = scroll's caster level + 1) to cast the spell successfully.

In this case: yes a first level wizard will be able to try with any arcane scroll (if he has the required Int) but he will have to perform an appropriate caster level with the following consequences:

If she fails, she must make a DC 5 Wisdom check to avoid a mishap (see Scroll Mishaps). A natural roll of 1 always fails, whatever the modifiers. Activating a scroll is a standard action (or the spell's casting time, whichever is longer) and it provokes attacks of opportunity exactly as casting a spell does.

Use magic device

You can use this skill to read a spell or to activate a magic item. Use Magic Device lets you use a magic item as if you had the spell ability or class features of another class, as if you were a different race, or as if you were of a different alignment.

This allow to override the first condition: even if you are not of the correct class (or have the wrong alignment and so on) and you have the use magic device skill you can do a check and try anyway.

So in this case if your 20th level fighter is trained in use magic device: yes he can use a scroll

I am largely uninterested in how similar results can be achieved in Pathfinder, so I've narrowed this to Dungeons and Dragons 3.5. I am also uninterested in changing forms, but I assume doing so adds significantly to these totals; I'll leave that exercise to another. This was exercise enough. Undoubtedly, there are bits and pieces here and there that can add to these totals, and already using haste, expeditious retreat, and the homebrew class are significant.

Dungeons and Dragons 3.5 feats that increase land speed

There are a lot. Like Sturgeon's Law says, most of them are terrible. Note that some of these feats were originally printed prior to the Dungeons and Dragons 3.5 revision and may be subject to minor adjustments by the DM (DMG 4). I'm pretty sure none grant enhancement bonuses so they should all be usable while the expeditious retreat spell's duration continues. The following list is alphabetical.

• The general feat Altered State (Dragon #332 84) allows a creature that takes 1d6 days to be dealt 1d4 points of Wisdom damage and make a level check (DC 20). Success means that the creature can take 10 min. to make a Handle Animal skill check (DC 20). Success means that for 1 day/level the creature gains a +10 ft. bonus to its land speed and a +2 bonus on Initiative checks. The DM may limit this feat's availability to barbarians.
• The domain feat Animal Devotion (Complete Champion 54-5) grants the creature at least 1/day the ability to take a swift action to gain for 1 min. a +5 ft. bonus to its speed that's +10 ft. at level 4, +15 ft. at level 8, +20 ft. at level 12, +25 ft. at level 16, and +30 ft. at level 20. Enjoy being covered in black spots. Gaining the ability to turn undead without taking cleric levels is a thing.
• The general feat Bind Vestige (Tome of Magic 72-3) and Improved Bind Vestige (TM 74) allow the creature to bind an inferior version of the vestige Ronove, that thereupon grants a +10 ft. bonus to the creature's base land speed.
• The wild feat Cheetah’s Speed (Complete Divine 79) grants the creature the ability to spend a use of wild shape to for 1 hour change its base land speed to 50 ft. Further, 1/hour, the creature during a charge can move at ten times its speed. A generous DM may allow this to be combined with the general feat Savage Mobility (Dragon #326 33) for another +10 ft. bonus to land speed. Gaining wild shape without taking druid levels is a thing.
• The general feat Dash (Complete Warrior 97) grants the creature a +5 ft. bonus to its speed.
• Probably something can be done with the tactical feat Faith Unswerving (Tome of Battle 34-5) and its tactical maneuver keep up the pressure, which allows a creature to take an immediate action before the creature's next turn and after a successful attack's made during a charge to move up to the creature's speed as long as the creature ends its movement next to the foe that was successfully attacked. This is awfully complicated, though. Probably better to stick with the feat Travel Devotion (Complete Champion 62-3) than try to work in tandem with a friendly demon with at-will greater teleport or whatever.
• Every other turn the tactical feat Perfect Clarity of Mind and Body (Tome of Battle 35), when the tactical maneuver coiled spring is employed, grants the creature a +30 ft. bonus to its land speed.
• The general feat Sprinter (Dragon #313 31-32) 3/day grants the creature the ability to gain a +10 ft. bonus to its speed for rounds equal to its Constitution bonus. One hopes the DM rules this is not an action to activate.
• If essentia can be rustled up, the general feat Shape Soulmeld (cerulean sandals (Magic of Incarnum 63)) (Magic of Incarnum 40) when so bound adds +5 ft. bonus/invested essentia to a creature's land speed. Other soulmelds have similar effects, but this one's earliest and available with the least (ahem) investment. Likewise, the similar feat Sapphire Sprint (MoI 40) if a creature somehow has more essentia than sense-ia. Thank you. Try the veal.

It's probably too difficult at this point to get the feats Cheetah Tribe Sprint (ShS 19), Divine Alacrity (Faiths of Eberron 145-6), Divine Vigor (Red Hand of Doom 126), Fire Feet (Dragon #311 40), Fury’s Focus (Dragon #304 84), Imbued Healing (CC 60) to use it with the domain Travel, Improved Power (Travel) (Dragon #342 21), Initiate of Fharlanghn (Dragon #342 51), Longstride Elite (Races of Eberron 114), Mark of Malbolge (Fiendish Codex II 84), Minor Shapeshift (Complete Mage 45), Prophecy’s Explorer (Magic of Eberron 50), Rage of the Raptor (Dragon #315 26), Speed of Thought (Psionics Handbook 29), Swiftness of Orien (Player's Guide to Eberron 48), Technomagic Implant (Dragon #351 93) for the cheetah's swiftness implant, and the trait Quick (Unearthed Arcana 89).

Unless shenanigans are used, Epic Speed (Epic Level Handbook 54) is likely unavailable, and it'll requires some serious persuasion to allow the psionic feat Psionic Boost (Expanding Your Mind Web column “New Talents, Feats, and More”).

Finally, it's probably too late to pick up the regional feat Fleet of Foot (Player's Guide to Faerûn 38), the dragon-only feat Improved Speed (Dr 71), or to be a saurian shifter with the shifter trait junglerunner (Dragon #328 63).

Dungeons and Dragons 3.5 magic items that increase land speed

I'm pretty sure none grant enhancement bonuses so they should all be usable while the expeditious retreat spell's duration continues. This list is also alphabetical.

• A creature with at least two air elemental grafts gains a +5 ft. bonus to its land speed per graft beyond the first while unarmored or lightly armored and not carrying a weighty load. Magic of Eberron presents only four different air grafts, two probably conflicting (one taking up the flesh slot and another the skin slot).

• It's a little difficult to find a magic item enabling consistent use of a personal spell by a non-caster, especially seeing as how the skill Use Magic Device isn't a class skill for the swordsage or the tenken. Assuming the creature's not devoting cross-class ranks to the skill Use Magic Device, a handful of mildly distasteful options remain: the feat Attune Gem (Magic of Faerûn 21) creates at potion prices plus 50 gp per spell level gems that the user can crush as a standard action to cast such spells, and the feat Craft Wondrous Item allows the creation at potion prices spellworms (Dragon #343 80-1), an internal spell storing parasite that allows a spell to be used either as a standard action like a spell-like ability or as a full-round action like a spell (the text is contradictory). More palatable—and pricey—is something like a ring of spell storing, but this might involve more than is comfortable a friendly caster. Below I've assumed spellworms (and a standard action) because they're hilariously gross.

  • A spellworm of footsteps of the divine modified by the feat Extend Spell [trans] (Complete Champion 120-1) (3rd-level spell modified to a 4th-level spell at caster level 20) (Dragon #343 80-1) (4,000 gp; 0 lbs.) allows the creature to take a standard action to gain for 40 rounds a +50 ft. bonus to land speed by picking Fharlanghn or, if discharged, to gain for 1 round a +10 ft. bonus to land speed per round remaining in the spell's duration. Presumably, discharging the spell isn't an action, but ask the DM. This could net in the neighborhood of a +400 ft. bonus to the creature's land speed for 1 round.
  • A spellworm of greater celerity [trans] (PH2 105) (8th-level spell at caster level 15) (Dragon #343 80-1) (6,000 gp; 0 lbs.) allows a creature to take an immediate action to take a full-round action at the expense of being dazed for its next turn. If the DM agrees that a magic item that stores a spell can be activated using the same action as the spell's casting time, this can get the creature another extra turn.

• The belt of battle (Magic Item Compendium 73) (12,000 gp; 0 lbs.), in addition to granting the wearer a +2 competence bonus on initiative checks, allows the wearer to take a swift action to spend 1, 2, or 3 charges to gain a move, standard, or full-round action, respectively.

• A creature with the spellware enhancement boosted reflexes (Dragon Annual #6 52-3) (13,500 gp; 0 lbs.) gains a +2 enhancement bonus on Initiative checks and a +10 ft. bonus to speed.
• The best are the horned helm (Arms and Equipment Guide 133) (38,000 gp; 1 lb.) that, among other effects, doubles the wearer's base speed, and the aforementioned magical shortspear rapid wrath (Ghostwalk 66) 11,702 gp; 3 lbs.) that, among other abilities, doubles the possessor's speed. These two items grant any creature a land speed of 120 ft.

Undoubtedly other sources of speed increases can dug up, but these are what I knew of off-hand. It's likely enough, but Dumpster-diving for even more is certainly a option.

"How fast is that?"

Using just the resources in this answer—excluding bonuses from expeditious retreat, haste, and the homebrew base class, for instance—, consistently, this isn't as fast as you want, but, in a burst that expends a lot of resources for only 1 round, it's way faster than you need.

• Turn 1: Take standard action to activate wild shape to activate the feat Cheetah's Speed. That's a base speed of 50 ft. which the horned helmet doubles to 100 ft.
• Turn 2: Take a swift action to activate the feat Animal Devotion, picking cheetah's sprint. That's another +15 ft. for 65 ft. Take a standard action to activate the spellworm of footsteps of the divine.

• Turn 3: Activate the feat Sprinter to gain for 1 turn a +10 ft. bonus to speed; this is not an action. Discharge the effect of the spell footsteps of the divine to gain for 1 round a +390 ft. bonus to speed (assuming an actual not relative duration); this is not an action. Take a full-round action to run. That's a speed of 515 ft. that's doubled via the magic weapon rapid wrath that's ×5 via the feat Run to 5,150 ft.

  Take a swift action to activate the belt of battle to gain an additional full-round action and take another full-round action to run. That's another 5,150 ft.

  Your turn should be over, but take an immediate action (spending the next turn's swift action) to activate the spellworm of greater celerity to take a third full-round action. On this full-round action, charge a foe, object, or square at ten times your speed via the feat Cheetah's Speed. That's another 10,300 ft. that's ×2 due to the maximum distance a creature can charge during a round.

This round, you've traveled 30,900 ft. in 6 seconds. That's about 3,500 mph, or over mach 4.5, which isn't even close to the speed of light but should be sufficient.

I don't want to get into Internet scuffles with anyone, so if it rubs you wrong to have activated the spellworm of greater celerity this way, skip that step and subtract 5,150 ft. from the total for, instead, about 2,900 mph or about a generous mach 4.

Note that Dungeons and Dragons 3.5 uses real-world multiplying for real-world things like speed (PH 304); this is slightly different in Pathfinder but not much, but, as always, ask the DM.