afaik, the research points you get for trading are a percentage of the points you spent on actual research (10% maybe - not sure). so if you're halfway through the civ tree, you spent maybe 2000 points worth of research, which creates about 200 tradeable civ research points. when you acquire research points from an AI, they are added to the appropriate tree. so if you buy 200 civ research, those points are added to your current civ research project.

well warfare knowledge gets added to your warfare research. i'm not sure how it's applied, though. i think it gets added to a random warfare tech that is currently available.

Yes, the main point being it is type specific. It's also a ratio you can mod in ElementalDefs, if you're so inclined = P

if they didn't use up their "knowledge" (i think that's the in-game term for the tradeable tech points) then it would be 10% (or something) of the whole research they invested in civic techs. for example, they spent 2000 research on civic techs and got ~200 knowledge for trade as a side product. if they trade away that tradeable knowledge, it's gone (this explains why a more powerful Ai only has 2/3/3 - they probably traded the other points away)

btw. you don't lose anything by trading it away - it's basically a 10% bonus on top of your spent research. i.e. if you research at 50 points/turn, you get 50 research towards your current project and ~5 points worth of trade knowledge that you can use as a trade good (exchange for other civs' research, gold or resources).

i think the government screen (foreign relations) has a statistic that tells you how much overall research they have accumulated over the course of the game. i don't use that screen too  often, so i can't tell more about this, but i think that's what the tech figure on that screen represents.

Yep, you get a bonus to the research tree you traded for, not your current research.