As the SRD says this:
Able to cast 2nd-level divine spells and 2nd-level arcane spells.
on the Mystic Theurge, would you be able to take the precocious apprentice feat to become a Mystic Theurge with one wizard and three cleric levels?

Or otherwise with a third level wizard who has the feat which allows him to gain the spells from a cleric domain?

Depends on your (or your GM's) reading. I've seen both these reasonings:

Yes, precocious apprentice is sufficient.

No, because the entry requirement states "able to cast 2nd level [X] spells" and precocious apprentice grants one 2nd level slot and one second level spell you can cast despite not having the required caster level.

However...
In summary: No, you don't have to make a full attack after the teleport - but if you used the Telflammar Shadowlord's Shadow Jump ability (or the Dimension Door spell), you can't take any other actions.

With other teleport abilities... you should be able to take free/non-action actions. I wonder, though, what kind of free/non-action you'd need between teleporting and full-attacking that you can't use before teleporting.

What if you teleport around a corner and discover someone you need to shoot that you couldn't see prior? Could you use that time to draw an arrow?

Does it specify it has to be a melee full attack?

Well, it seems no one is adept with 40k rules, so I'll just ask a 3.5 one;

Can a druid quality for the Alienist PrC and Summon Nature's Ally quality as Summon Monster?

The shadowlord must have line of sight to his intended target prior the teleportation, so attacking a 'new' enemy with shadow pounce is out.

Yes and no.

A Shadowlord with reach holding a dagger could teleport into melee distance and then make a ranged full-attack with thrown daggers (assuming Quick Draw and a teleportation effect that's not based upon Dimension Door, i.e. no Teflammar Shadow Jump)

Qualify? Yes. Summon Nature's Ally is a Conjuration (Summoning) Spell.

SMN, however, would not benefit from the Alienist Summon Alien class feature without houseruling; Summon Alien specifies summon monster spells and without houseruling, this excludes Summon Nature's Ally, Summon Undead, etc.

Another question:

I know for a fact that there is a feat allowing you to use magical items from a class as if you belonged to it, but I can't for the life of me find it. Would any of you guys know what feat it is?

A simple UMD check allows you to do this.