The fourth major thing that spies do is steal technology. I've had spies bring me as many as 30+ techs over the course of a long game (admittedly, that was a spike in performance, but 15+ is still nice.)
Part of the secret is that you seem to need to get High or Advanced Intelligence level on a civ before stealing gets to a useful probability. So what I do is hit each civ I encounter with a series of spies til I get Low Intell, then switch any spies beyond 1/per civ to the next civ. When the one that is left behind at this stage attrits, I just let it go for that civ for awhile.
After I have Low on every civ I can reach, I focus spying on one civ at a time until I get Advanced, or at least High, on each one in turn. Just keep plugging them in there. Surprisingly often (but less than half the time), a target civ will seem to sort of accept the situation after awhile and stop policing so aggressively, so you will get up to 3-4 spies on a civ. Once I reach Advanced, or High if I have 2+ spies on that civ, I move the focus of my spying to the next civ, while attempting to maintain at least 1 spy on the civs that I have good knowledge of.
If you are trying to limit the activities of a civ, I suggest nailing any money or population improvements. The AIs always play "calculatingly" close to their resource limits, so sudden shortages in those two areas can have at least minor cascade effects. If you are trying to NOT hurt an ally, while building your knowledge base on them, I suggest spying on (ta-da) research facilities; it actually hurts them the least, and seems to annoy them the least. After all, allies can trade with you for their techs.
Oh, and as far as I can tell, the AI is not any more able to determine who is dumping all those spies on them than you are. I've never been able to notice any increase in an AI characters "ticked off" factor when I've been doing extensive spying on them, not even the time I had 1.5 planets worth of spies on the !! Korath's !! 2 planets. (Stole him blind, though!)
I know that there are not a lot of spying fans on these forums, but I like to play ideally bloodless, conniving strategies, going for alliance or influence or fully-teched-out tech wins, so I hope that helps.
Spying could use a couple of good governor options, however, to target them and maintain the spy pressure on a civ without having to click-place-replace each spy (Frogboy - hint, hint).
drrider