1) Is considered flat-footed an opponent shooted by a crossbowman with improved deadshot for the purpose of deadly stroke?
No, being Flat-footed does deny dex, but they are not the same thing. It's kind of like asking if having jelly inside makes something a donut.
2) What it's doubled with deadly stroke? Weapon magical bonus? d6s from magical properties? Wounding? The other d8 from the double crossbow?
You double all the normal damage--i.e. not precision damage like Sneak Attack. So, yes to Magical Bonuses, yes to magical properties. Wounding deals bleed, so no. The other d8 is not normal, that's a special property of the double xbow, so no.
3) One-handed firearms are considered light for the purpose of two weapon fighting?
I've not seen anything like that, but I am not really "up" on the firearms rules, so it could be in there somewhere.
4) Must I add bonuses from amulet of mighty fists when I'm using awesome blow?
You certainly don't have to, but why would you not want to?
5) Why the vivisectionist (alchemist archetype) have in discoveries some bombs even if he can't use them?
Because the discovery list doesn't change for different archetypes. Just don't take bomb discoveries, obviously.