Let's see, off the top of my head.
1) "Show enemy movements" allows you to see if a stack is "grouped" or has a straggler or two. If there is a stragler, then the stack can attack and die and the straggler can retreat.
2) Cities can be gifted to another civ if your gift city is within 9 tiles of their closest city. Diagnal tiles are counted as 1.5, and any 0.5's left over get truncated.
3) You can (Liberate) a city to another civ if it is within 9 tiles of their capital and you haven't generated any culture in it yet.
4) If you and a hated civ are at war together against a 3rd civ, you can open borders with the hated civ no matter what the relations are at.
5) The open borders through shared war can be abused to teleport units. This is because canceling an open border results in "not right now, maybe in a few years" before they will sign a new one, which is like 25 turns for most civs, but with shared war this too is bypassed. So close borders to teleport, then sign a new open borders treaty right away if there is a shared war with the civ.
6) Trade Embargos are a perfect way to get two civs to stop having open borders or liberating cities to each other
. They simply refuse to talk to each other for 20-30 turns I think.
7) DanF is an awesome team advisor. He has the combined Strategy Forum's knowledge from 2003-2033 in his head.