Is it possible to get all of the unarmed moves npcs can teach you (Khan Trick, Scribe Counter, Legion Assault, Ranger Takedown, and a standing power attack from House?) and if so, how is it done and in what order should I do it? I fully intend on siding with whatever faction requires the most quests completed to get their unarmed move. Also, does House really teach you an unarmed move? It says so on the wiki but there is no confirmation.

Yes. but Khan Trick kinda limited the way you attacks and speech.

OT a bit, Khan quests are pretty badly designed. In fact, I think it is the only faction which you can't join, nor offers follower and safehouse.

I plan on having a high Speech if that's the problem you're referring to. Can I have a neutral standing with the Khans and still start "Aba Daba Honeymoon?"

I think so.

But cut the option of side step punching make it pretty crap IMO.

Oh, so the Khan Trick is actually worse than the normal sideways attack?