I'd say that the feat grants a Magical Training spell list, not a Sor/Wiz spell list.
When you take the feat, you get to pick 2-3 spells from the Sor/Wiz spell list, and those spells make up your spell list. You need to be an actual Sor/Wiz to gain their spell list.
Likewise, you're only considered a Sor/Wiz with regard to those selected spells, and for nothing else. So I'd say that you don't get to use items that require Sor/Wiz as the classes, but you still pass as an Arcane Spellcaster because those selected spells are arcane.
This is why the feat can pass the Dweomerkeeper's requirement (arcane spellcasting), but not exactly for the wands that you mentioned.
Casting spells is different from knowing/preparing spells. The only connection is that when you cast a spell, it must be from those you know or have prepared (with limited exceptions for some rituals). There are 3 tables for how many spellslots a character has to cast with -- 1 for full spellcasters (wizard, cleric, bard, druid, sorcerer), 1 for half casters (rangers and paladins), and 1 for third casters (eldritch knights and arcane tricksters). Warlocks have a different system for this.
Know or preparing spells is different.
Wizards, clerics, druids, and paladins prepare spells -- each has a number of spells they can prepare, and can change them after a long rest. On that day, they can only cast those spells, using their spellslots to do so. Clerics, druids, and paladins choose the spells they have prepared for the day from the entire list of spells; wizards choose from those they have in their spellbooks.
Bards, rangers, warlocks, sorcerers, arcane tricksters, and eldritch knights are much more limited in spell selection. For each of these classes, in their class table it says how many spells the character may know at any given level. When they go up a level, they check that table to see if the number of spells they know goes up; if so, they pick a new spell (or spells if it goes up by more than one) to add. At leveling up, they also have the option to change one spell they know to a new one that they can cast.
When the spellcaster casts a spell, this does not in any way affect which spells he knows or has prepared. Those do not change with casting; each time he casts a spell, he can cast any spell that he knows or has prepared, using a spell slot equal to or greater than the level of the spell he chooses to cast.
So, to answer your request for examples:
A 5th level ranger knows 4 spells. He has just gained the ability to cast 2nd level spells, so he can at most know 2 of those (one for the spell he gained at 5th level, and one could be changed from one of the 1st level spells he knew previously). The other 2 spells he knows will be 1st level spells. He has 4 1st level and 2 2nd level spell slots with which to cast those spells he knows.
A 5th level paladin can prepare a number of spells equal to his charisma modifier, plus 2 for his level. So, for example, if the paladin has a 16 charisma, his modifier is +3, and so at 5th level he can prepare 5 spells. After each long rest, he either keeps the spells he has prepared, or spends a few minutes preparing a different set, from the paladin spell list. He can prepare 1st and 2nd level spells (since that is what he is able to cast), in any combination. He has 4 1st level and 2 2nd level spell slots with which to cast those spells he has prepared.
Best Answer
If it's Disintegrate? No, you still can't cast it.
Distintegrate's casting requirements are the following:
That's as per this rule:
The feat Eschew Materials would remove the Material requirement, at which point the spell could be cast.
A spell that did not have a M component in its stat block wouldn't have that issue, and you could use just Still Spell to cast it.
Can you Aim It With Your Hands Tied?
Distintegrate's text says this:
Still Spell removes the Somatic component of casting the spell, but nothing in it's text says that it removes the need to aim the spell or the mechanism by which you do that. So by RAW, you would still have to use your finger to aim the spell, with your hands tied.
If your hands are tied in front of you, it wouldn't be that hard. If your hands are tied behind your back, it's going to be pretty tricky as you can't see what you're pointing at. In that case I'd use the rules for Total Concealment (50% chance to miss whatever you're aiming at). This raises the issue of figuring out how you're tied up, which is quite possibly something that wasn't thought about if nobody described it and a Use Rope skill check was used to do it.
The other interpretation is that the finger pointing is just part of the Somatic component (hand gestures) required to cast the spell. If you interpret it such that it is, then Still Spell also removes the pointing requirement, and you would aim some other way. Probably by eye contact. You'd still have to make a ranged touch attack, as per the spell description.
In either case, penalties for being tied up might apply depending on how well you're tied (a fully bound character is Helpless and has an effective Dexterity of 0, which is a -5 on the attack roll). If they just secured your hands, a lower or perhaps no penalty would apply, that would be a DM judgement.