I unlocked a next tier award today and it gave me a building that requires 5 political favor to build. I was wondering what is political favor - and how do I gain it.

its for building certain structures, your gain it from completing/continuing certain missions.

Is there any structores other than soldier outposts that require politic favor? How much politic favor per structure (including soldier outposts)?

Yes, (only 2 that I know of though) pfft unlike most of the people here I dont either give away EVERYTHING or do a sissy job of hiding it. I will tell you though that its nothing thats going to make spend huge amounts of time/energy to get, its just big enough that they arnt being given away. (Someone has to leave something for others to find out or be surprised by)

Understandable.

I was just concerned because missions poof when you complete them, so I was making sure that I'll be able to have enough to build what I need on good planets and stuff.

ya if you fully complete the mission you should have enough to make around 8 of them, and after a little bit theres another mission that gives even more.