I'm pretty sure that the active treaties isn't giving all of the information at present.  In my current game, it's late-game, and almost all the civilizations have active research and economic treaties going somewhere (made clear by the fact that they are unavailable for me to negotiate for).  But the active treaties screen is showing only a few of them.  I assume there's some espionage level requirement to know who has active economic/research treaties with whom.  If not, then there might be a bug in that regard.

As a separate issue, from what I've seen the AI never tries to negotiate research or economic treaties with the player, does it?  Why not?

The Cap'n's right about looking at the View Report list in the Trade screen--just click Treaties Report. You can step through everyone who'll speak to you at the moment by clicking the arrows at the bottom corners of the viewscreen area.

I've always sort of thought that was because I tend to have low popularity until late in a typical game. If this is true, it seems like a violation of the general guideline that the computer players follow the same rules as the human. They certainly swap those treaties freely amongst themselves, but if they never ask the human player, how (why) are they asking everyone else, including the minors?