migthegreek said:
Do open borders make your relations better with a civ? (and how much so)
Yes. Like supplying resources and peace, it builds up over time, in general won't get better then +2 though.
migthegreek said:
Does the AI start garrisoning loads of units around your territory (like they did in Age of Empires, if anyone knows)
At this point, it is clear that you
don't know that on declaration of war, all units of that side get kicked out (the same happens if you cancel Open Borders without declaring war), which stops what was known in Civ III as "ROP rape" (equivalent of open borders in CivIII was called Right of Passage). Of course back then, it was far more often humans doing it to the AI not the other way around

.
Thus the only downside of open borders is that they can scout out the disposition of your defences and move through your land to empty lands beyond. And that last point is pretty much the only important one, if them scouting you out reveals you to be weak enough to be worth attacking, then your defences weren't good enough in the first place

.
Thus if I have cut of a peninsula that I plan on settling or similar, I may refuse all open border requests, but beyond that I'll freely sign open borders with anyone I want to be friends with. The improvement to relationships is small but nice, but more important are the foreign trade routes.
As for the list of 3, I don't really see how the distinction between the first 2 can be made even in historical/real world terms. How would you know whether those foreign merchants are really scouts/missionaries/great people? Missionaries can effectively be shut down with Theocracy if neccessary, and even without actual missionaries turning up religions can spread (subject to Theocracy) with luck anyway. Once they get to scout your cities, it really doesn't matter whether those use scounts or armies due to armies getting kicked out at the start of hostilities, so in reality there's no tangible benefit in complicating the single open borders agreement system in the game.