Having read the manual I'm still a little unclear on how religion works. Here's my questions-

1. If I have founded a religion and introduced it to another civ I will get payment from that city?

2. What if they have other religions set up in the same city, do I get less payment?

3. Do I only get payment if I have personally introduced that religion to that particular city? What if I founded it but someone discovers it themselves later on and introduces it themselves?

in order to gain payment, you need to build the religion's wonder in the founding city, which you can do using a great prophet. after you build it, you gain +1 gold for every city that has the religion. it doesnt matter if you send a missionary to spread the religion, or if it spreads alone. you still get your +1 gold from cities with more than one religion.

Thanks. Is there anyway I can check the income from my temples? Will in be listed in the trade route panel?

you enter the city view of the "holy city" in which you built the religion's wonder, and look for the woder on the left buildings panel. you will see a +number 'gold' next to it.

Thanks. Is there anyway I can check the income from my temples?
Your temples only produce money if you build the Spiral Minaret, and then only the temples of your State religion produce money. As BuFu mentioned, you can see the income generated by any of your buildings by opening up the city's details (by double-clicking on the city.)

Which also means that any Boni you get only for buildings of your state religion (like the money gained from the spiral miaret) disappear as soon as you use Free Religion as a civic (as with free religion you don´t have a state religion anymore)

... which is why I don't go for Free Religion until after I've got Computers, which makes the Spiral Minaret Obsolete.

It also says you can investigate the cities you have introduced a religion to but I can't seem to be able to(??)

By "investigate" it means that you can "see" the city's fat cross (it is lit up and you can see units)

as long as you don´t have Beyond the sword installed,
where you don´t have this ability anymore ;)