So my question here (which is open ended, so thus no poll), how do you handle which gods are available in your game?
I have a lot of them. My homebrew setting is called the Dome of Heaven, after all. There are four elemental deities, who rarely get involved in human affairs. There are the Regents of the Dead, who rule the lands of the dead. There are the Royal Deities, who are the most well known and the most powerful. There are the Noble Deities who are lesser known/powerful, but still recognized throughout the (huge) world. Then there are the common deities, of which there are a ton. My player hand out has as many common deities as all the rest combined, and if the player wants something specific that's not on the list, I would work with them to create a common deity for them. I have always loved the idea of the gods in Lankhmar, as opposed to the gods of Lankhmar, with a long street of worshipers for god after god after god...
There are also some non-deity centered faiths. There is animism, which is based on the spirits in the Astral Sphere that reflect animals, plants, and geography. There is Moism, a more philosophical way of life founded by a semi-mythical mercenary named Mo.
Do you pick a pantheon for each kind of people (elves, humans, dwarves)?
I don't like this and tend to stay away from it. There are deities who are associated with certain species, and look like one of them. But the idea of religion in my world is that people go to different deities at different times based on what they need. When starting out they might make offerings to Alodar, the god of luck and wealth. When they meet that special someone they might seek help from the priests of Iosephine, the goddess of love and truth. When the wife becomes pregnant they will of course pray to Mah, the goddess of fertility. Later they may pray to Obi, the god of farming an family. If their son is killed, they might make a dark vow with Morcant, the god of vengeance.
Do you pick pantheons for regions?
It's possible. My world is a big place, and some regions may have particular (common) deities specific to them. It hasn't come up yet, though.
Do you use real-world pantheons, D&D setting ones, or totally homebrewed ones?
I have several real world deities among my list, like Thor, Tyr, Asmodeus, Anubis, Hexate, Artemis, and Oghma. But not whole pantheons. I used to pull from the D&D ones as well, but after the OGL debacle I went through and trimmed out any WotC specific IP from my world, just to be on the safe side.
Does this only matter if there are cleric PCs?
Not really. Moism could be anyone. Animism is common among druids, but there are various nature related gods they might be devoted to. I don't use alignment in my games, so instead I have core tenets for each deity. Clerics are expected to have a deity, and to follow the core tenets of that deity. In extreme cases they could lose their clerical powers by pissing off their deity, but I would allow them to change deities or atone to get their powers back (Hey! Side quest!).
Anything else you think is relavant?
My view on world building is that I like to do it, so there is a fair amount of it out there. But I don't try to insist that my players enjoy it. I have a 43 page document about my homebrew world, but I don't expect the players to read it. It's there if they want it, and some do. When things come up in the game that depend on my world building, I let them know what their characters would know about it, maybe after some history or religion checks for the more obscure things. When they ran into some locathah who were trying to rebuild a statue of their god to protect them from a giant, psychic lobster, it was an easy religion check to recognize Harro, the god of rivers and death.
In terms of interacting with the world, most of what deities do is through their worshipers. Right now my players are dealing with a nasty cult of a dark aspect of Chippick, the god of rats and deserts. As they get to higher levels I expect more interaction with celestials and fiends sent by the deities. Dealing directly with the god would be at much higher levels, although they might get visions or messages at the lower levels.