I would expect that a Religious city-state provides Faith to its friends and ally in the same way as a Cultured city-state provides Culture.
Some of the Beliefs (such as Papal Primacy) specifically mention city-states following a religion, which implies to me that city-state follow world religions, rather than having their own. Since having a different religion imposes a diplomatic penalty, it wouldn't make much sense for Religious city-states to have their own religions, because then by definition they're going to have a different religion than you. So, I would guess that even Religious city-states start with no religion (or perhaps a Pantheon), and convert to another religion normally.
As to how religion affects diplomacy, I'm going to guess that having the same or different religion just adds an influence bonus or penalty. But that's a guess -- it could be more complicated.
There was a mysterious civilian unit that belonged to a city-state seen during a PAX demo. It makes me wonder if religious city-states can produce missionaries.
That would be my guess, but if so, it is an intriguing question of why a city-state would want to produce Missionaries. The owner of the unit in question was Brussels, which is (as far as we know) a Cultured city-state.
but otherwise it was a waste of time in Civ 5. so maybe they enhanced it as well.
This is one of those cases where the feature is useless unless the AI is smart enough to respond properly. There was nothing wrong with the "pledge to protect" mechanism except that the pledge did not appear to have any effect on the actions of the AI civilizations. When the AI made the pledge, it did seem to come with a real diplomatic penalty as a consequence for attacking a city-state, but I almost never attack city-states, so I totally ignored these pledges. The advantage of the new "threaten" system is to offer a real reason to mess with city-states (but not conquer them, which is counter-productive), so that pledges may be meaningful. But they still will have problems if the AI's don't respond properly to them.