I know that if I build the Temple of Solomon, for example, I get coin in that city for every other city that has Judaism. But where do those coins really come from? Am I taking them from my other Judaism cities? Am I robbing Peter to pay Paul, as the saying goes? Conversely, if another civ has the Temple of Solomon, is he taking coin from my Judaism cities? Could that be a reason to keep other religions out of your cities?
Or do these coins just appear out of nowhere, sort of a spontaneous output from the citizen's own pockets?

The coin appears out of nowhere.

The coin appears in your wonder city but is not subtracted from anywhere else.

Donations from pilgrims...

I thought you had to build a temple (of whatever religion) in order to get a coin from the shrine (of that religion), or do you just need to have a successful missionary attempt bring that particular religion to a city (with no need of any religious buildings built thereafter)?

A related question: I've noticed that sometimes a foreign country will spread a religion to one of my cities, but the game won't let me build any missionaries of that religion (even though I build a temple & mission in that city). I don't know why it won't let me build any missionaries in that case?

Temple gives you +1 happy face :), which looks like a coin of commerce :commerce: (except with eyes and mouth), +1 culture, and the ability to work one priest specialist. You don't actually get commerce unless the happy face made an unhappy pop productive.

The shrine (built from Great Prophet) gives you +1 gold :gold: from every single city (including the holy city itself) that has that shrine's religion in it, regardless of what other buildings exist, regardless of which civ the cities are found, and regardless of what other religions are resident in the cities. Pilgrims are profitable. That's gold, not commerce, BTW.

Building the monastery only gives you the ability to build missionaries, +2 culture, and +10% science :science:. Monasteries go obsolete so you will want to build it before they do. Otherwise, you will have to spend some time with the Organized religion civic.

If you already have three missionaries built or building in another city of that type, you can't build more. If I can't build a missionary even though I have the monastery, it usually means that it is already on the build queue three times in another city, or they are already built and still heading for the cities where I intend to use them.

Not sure if by Mission you mean Monastery, but you need a Monastery of the 'desired' faith before you can create a Missionary. The exception to this is when you are running the Organized Religion civic.

There is also a 3 Missionary limit.

EDIT: Beat me!

The religious building is actually good for makign a great prophet factory. When u are short on time and dont want to bother building 3 temples, pop one and you can make 3 priests in that city! Of course I know it takes one Great Prophet but it can pay off in long run if you are a phi.