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

Does all this means that a faerie dragon

1. can cast scrolls and wands with spells from the sorcerer/wizard list without doing a UMD check?

Yes. Wands can be used automatically with no difficulties whatsoever; scrolls work as the faerie dragon either has a high enough caster level for the spell, or succeeds on a (relatively easy) caster level check.

2. has to perform an UMD check for all other spells not on the list?

Correct.

What does [“the user must have the requisite ability score”] mean for scroll and wand activated spells from the sorcerer/wizard list?

This... is actually somewhat contentious. Spellcasters require a certain ability score in order to cast a spell, e.g.

To learn or cast a spell, a sorcerer must have a Charisma score equal to at least 10 + the spell level.

To learn, prepare, or cast a spell, the wizard must have an Intelligence score equal to at least 10 + the spell level.

The problem is that it’s unclear which requisite ability score is relevant. If a wizard wants to cast a 3rd-level spell from a scroll scribed by another wizard, he needs Int 13. But if a sorcerer (or faerie dragon) wants to cast that same scroll, does he need Int 13 (which would be the requisite ability score for the wizard who originally scribed the scroll) or Cha 13 (which would be the requisite ability score for the sorcerer to cast the spell himself)?

Summoners get even more confusing, since they have spells at different levels: does a wizard-scribed scroll of haste require Int 12, Int 13, Cha 12, or Cha 13? After all, summoners have haste as a 2nd-level spell, so they require Cha 12 to cast it themselves, but it was still scribed as a 3rd-level spell by a wizard who needs Int 13 to cast it.

The general consensus, from a quick Google search, seems to be that you need the ability score that you would need to cast the spell yourself, not whatever the person who scribed the spell would have needed. The primary reasoning for this is that in all published adventures, you see loot like scroll of haste, not scroll of haste scribed by a wizard or whatever. On the other hand, there is a rule to assume cleric, druid, or wizard made magic items that don’t say, so that argument is not all that strong. It’s just the best we have.