I'm running Dark Sun, so ... no gods. Some pretend ones though.
If I were doing my own settings, though, I'd do the following:
I am curious on how different DMs handle and use the gods in their games. Can you kill them, do they play a major part in your world? Do you just hand wave them away except for letting the cleric have access to divine magic?
You can't kill them (or even get to them). They don't play a major role in the world. High-ranking divine casters and "immortals" play a role. The casters are often overzealous and the deities cannot tell them this. (A caster who uses the appropriate rituals might get a hint of this though.)
Do your good temples all get along is the only strife if there is any between good and evil churches? Do you require your clerics to tithe to their churches? Do the cleric gods every get angry with the cleric and block access to his spells?
1) Usually I only have a few religions. Often they're given very different portfolios, keeping conflict minimized. (Of course they compete.) Even differently-aligned deities might not be in conflict, depending on their portfolios.
2) Not when adventuring. Tithing depends on the setting. For instance, if I'm using a Thrane-inspired setting, there is a state religion, and everyone is expected to tithe. It's the law. The non-religious rulers still pay tithes, as they have confessors, who know everything.
On the other hand, if the setting is a bit more like Japan, there might be numerous religions, each of which competes for donations. Some people wouldn't pay anything at all.
3) Rarely. You don't have to use spells to further the deity's agenda (at least, not each individual spell) but using magic directly against the deity's agenda would get you in trouble.
Gods get power from their worshipers so there is always some rivalry between them even the good gods.
Gods need prayer, badly. Yeah, I like this one.
As for the churches sometimes they work together sometimes they don't. And if you are not a follower of the god you can't expect them to help you for free there will always be something in for them.
I agree with this, but I don't think it has much effect on PCs as they usually have a cleric or other divine character to rely on anyway.
Take raise dead the cleric preforming it on a non follower will usually be tasked by his god to do something or sometimes the god may put a geas on the non believer to fulfill a task or give them X amount of time to fulfill a quest or the god will take back their life. Which is why in a lot of my games people prefer reincarnate usually because it has no strings involved.
I don't like deities having this much direct influence on the setting. Unless you're trying to raise an enemy, there shouldn't be a backlash.
One question, primarily to those who tie the power of a god to its number of followers, what is a follower exactly?
In a polytheistic pantheon, especially one where people do know for fact that all the deities really exist like in D&D hardly anyone would just worship one god. They might slightly favor one god over others because of their profession, tradition, birth, etc. but most people would follow the rites of several, if not all, gods and pray to whatever god their support they need of think they have angered.
In Japan, you can be Buddhist, Shinto, both, or neither. I figure a character can easily have a "split loyalty" and is therefore only giving a partial amount to each religion, both in terms of money and in terms of "unspecified power". (In Ancient Japan, you were expected to follow whatever religion your daimyo followed. If you were Shinto, and your daimyo was Buddhist, you would be expected to be both. I'm not sure what punishment, if any, there was for not following that tradition.)
I don't like the idea of having characters worship "outside of their pantheon" and said characters would know that any curses, etc, would come from immortals or priests rather than the gods themselves. (If you ticked off a god, you could negotiate with their priests in order to not get cursed.)
However, a character who wanted a specific ritual performed (one for a business blessing, for instance, or a sea journey) could pass money to a follower of that deity and have them perform the ritual. While supporting a "rival church", this has no real impact on their own religious feelings.
Since most spells and rituals are not deity-specific, it's easy for a character to literally not need another deity for anything. Pretty much any cleric can cast Water Walk, even if they're worshipping a god of the desert. (It's not a spell they're likely to prepare every day, or a ritual they're likely to have on them, but they could prep or buy a scroll.)
As a result, supporting a rival church, even financially, is rare in a game I'm running.
Note that outside of D&D many more spells would be segregated. In Warhammer, for instance, Gentle Repose (I don't know what it's called there) would only be available to priests of Morr, and a priest of Sigmar might have to pay a priest of Morr to keep a dead friend preserved or prevent them from rising as an undead.
