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I have some questions about running a Deathwing army. Are they legal/GW official? Do they have their own codex-type thingy? Muchos gracias.

Codex Dark Angels

RaegMachine is correct, but I'll try to elaborate on his answer just a bit.

There is a special character (Belial) in the codex that lets you choose Terminators as both Elites and Troops. He's basically a 130 points Terminator Captain with Storm Bolter and master-crafted power sword (or twin LC).

Do they have to be Dark Angels, though? Can I paint them as Imperial Fists, for instance, and just use a generic Captain in power armour to count as Belial? My idea was to run an Imperial Fist 1st Company with Lysander as HQ

As long as you are clear with your opponent what's going on, then you should be okay. In a tournament, you may have to model your 'hero' to fit Belial and not Lysander.

"my blue space marines are a subchapter of dark angels using the dark angels list."

as long as the models are wysiwyg it should not be a problem.

As long as you use the DA codex and ONLY the DA codex. Assuming you model correctly there shouldn't be any problems.