What is the actual loss/gain due to defying a resolution?
Full members lose full member status - become voting members. Voting members can't become leaders of the AP
I believe there's also an unhappiness penalty in all cities that share the AP religion.
What are the triggers that allow various options to pop up ?
Don't know.
Well, every civilization needs at least one city with the religion for victory to be an option. I think a member needs to be at war with a non-member for holy war to be an option. If 2 or more members are at war you can vote for the 'end the war' against 1 of the members. There's also resolutions to sign open borders with everyone and for everyone to sign defensive pacts (probably only if everyone is at peace). You can also have trade embargoes against non members.
What happens if you own the AP and you switch religions?
That was my question. Edit: The AP will still retain it's original religion, but my question was whether or not the owner would be eligible for AP elections now that they're running a different state religion.
The AP always retains its religion, but the owner is always eligible for election, even if he has no religion or a different relgion.
What happens if you take the AP from someone else and they built it as another religion?
It's a world wonder - they can't rebuild it and the original religion the AP represents cannot be changed.
as above, you should be eligible for elections.
To stop a diplomatic victory:
1) Spread the religion among your own cities and to give yourself voting power (go to organized religion if you need to)
or
2) Spread the religion among the cities of the worst enemy of whoever leads the AP to ensure at least one AI won't vote for him
or
3) Conquer a lot of the cities with the religion to give yourself voting power
or
4) Raze any cities you capture that have the religion so you'll never get the religion
or
5) Bribe the two top civs in voting power to war with each other/ trade embargo each other. If they hate each other, neither will cast votes for the other for diplomatic victory
or
6) Adopt theocracy as a civic before the religion spreads to your civ
or
7) Get an ally without the religion to adopt theocracy
or
8) If you have vassals with voting power switch to the AP religion yourself so your vassals will be forced to vote for you instead of your rival
or
9) If your only cities with the religion are on another continent, spin it off as a colony, or liberate them to a vassal so you don't have the religion any more.
or
10) If you have cities on another continent or island without the religion, spin them off as a colony so there's at least one civilization without the religion.
or
11) raze the city with the AP...I'm not sure if this works, but others have recommended it.
There's probably some other strategies for avoiding an AP victory, but these are the only ones I came up with in the 10 minutes or so it took me to type this. There's nothing 'cheesy' about a victory you can avoid if you pay a little bit of attention.
Go to the victory screen (F8?) regularly and check out 'members' to see what religion the AP is, who owns it, who the members are, and what they're voting power is. You should be doing this regularly (and also check if someone is going for a cultural victory).