5) What are the priests like?

It’s certainly possible to invent a fantasy religion without priests, though I don’t think I’ve seen one yet; having a system of specialized attendants to the gods seems to be the default. It happened even with Protestantism in our world, where a large part of the original draw was the idea that people could interpret the Word of God on their own, without a church hierarchy interfering. If you have creatures with human-like psychology, it’s not beyond the realm of possibility that they’ll want priests, shamans, Vestal virgins, or something similar.
So, all right, you have your priests. How do they relate to the laity? What are their duties? How do they stay in power? What restrictions are on them, and what ones can they not break, and what ones can they slide past, and what ones do they merrily break all the time, perhaps with the gods’ full approval?
I think the last set of questions is especially valuable for people who tend to write priests with the Basic Priest Personality, which is a cousin of the Basic Healer Personality. Priests are kind, gentle, unworldly (although most of the time not actually ascetic) and yet wise,always faithful, and disdainful of violence. Priests of war-gods are damn rare in fantasy. So are priests who have their culture’s normal attitudes towards sex. It’s either ignored for them or treated as something exclusively sacred; I get the impression that most Mother Goddess priestesses don’t actually have sex outside the ritualistic surroundings prescribed for them to do so. If those attitudes spring directly from the religion, fine, but there are evil High Priests and High Priestesses all the time (also with stock personalities) whose main evil is often sexual. Yo, fantasy authors: not every culture will have a Puritan background. If your priests are sworn to chastity, or all the good ones have a horror of anything other than strictly for-children sex in the missionary position, please explain why. The complex of attitudes relating to sex in the Christian church is, well, complex, and not easily transferred whole to a fantasy setting.
Priests are some of the hardest fantasy characters to write because they aren’t often the main characters. Authors who can handle hundreds of personality variations for mages and fighters seem to get nervous when confronted with priests, and so turn them into perfectly good or perfectly evil people that the main characters of the story talk at and are talked at by. They’re Plot Devices. They shouldn’t be. Remember that their actions may well be the main lens through which “ordinary people” see the church, or churches. Consider that, and adjust accordingly.

6) What are the main doctrines of the religion?

Do you like developing holy days, calendars, religious history, and so on? The doctrines will be the reason for this. Why does this village celebrate the return of the Spring Goddess on the equinox instead of another day? Why is this shrine clipped and tended and never allowed to be overgrown, while this other one is completely ignored? Why is that high priest remembered as good and kind despite being an abrasive personality in life? Tracing the theology of your religion can be a lot of fun, and can give you ideas about what factions and heresies and even new religions will start because of it.
What is the decision-making process that your church uses about such things? Depending on how wide-ranging and democratic it is, it can change the whole future history of your world. The Bible as we know it, for example, leaves out a lot of books, including some Gospels, while including some that other people argued should be left out. The canon involved hammering, compromise between various factions, and savage suppression of heretics who decided they didn’t want to accept that canon, thank you very much. Priests were the ones who made the decisions, but the common people weren’t always happy with that. Perhaps in a fantasy world, that’s enough to spark off a rebellion, and the rebels win or at least hold their own.
That doesn’t have to be the case in your fantasy world at all, of course. Say the church is specifically based on an activity that is mostly accomplished by peasants. Farming would be the classic choice, but if there’s another activity that high nobles disdain and so is firmly in the hands of another class, that would put a beating heart of power in a place other than the palaces. Symbols can influence people’s minds just as well, and sometimes better, than temporal power—and of course it’s even better when the symbols and the temporal power are allied. The nobles try to declare a canon of the Banker God’s books, and the merchants themselves show up, wave around the money they got from successful following of the BG’s principles, and influence people to choose a different way. If the merchants have the BG performing miracles in support of them, or magic to make people think the BG is performing miracles, then they might easily take over power completely. This is a somewhat silly example, but it can be elaborated.
Consider holy days, too. What’s at stake when people choose a particular day to worship a certain god, or an event in the life of a certain god? It’s easy to say “Nothing” and be arbitrary about it (which is often the feeling I get, looking at a lot of fantasy religious calendars), or to insist that every holy day be tied to a solar or lunar event. But fantasy writing shouldn’t be easy.
Perhaps one village already has its own ceremonies in the middle of summer, in praise of a minor god whose worship is dying, and wants the holy day to be the same day as that one so they don’t have to give up their worship completely. Perhaps another can afford to put on pageants in the fall, with money come in from the harvest, but not in the winter, so they really don’t want a ceremony on the darkest day of the year, thanks. Perhaps a city hopes to lure more people to it in the summer than in the winter. Perhaps a king has chosen to identify himself with the god, so wouldn’t it make sense if the god’s birthday was also his own? All of these might seem overly practical, but considerations of doctrine—as opposed to the core principles from question 1, which are usually said to be dictated by the deity itself—often boil down to that.

7) How do the church and state work together?

Freedom of religion is an admirable principle, but once again it gets teleported into fantasy worlds with a history that doesn’t support it. If your peoples have a long and violent history of religious conflict, why do they tiptoe smiling around a stranger’s right to slaughter cattle in the marketplace, particularly if their religion thinks cows are sacred? The government would have to be constantly vigilant to enforce freedom of worship in such an environment, and the stranger’s nation would have to have tempting prizes to make it worth the bother.
Of course, perhaps the church and state are the same thing as one another. That’s possible. It still causes more problems than most fantasy authors are willing to acknowledge. What happens to heretics? Are they tried as solely religious criminals, or is heresy also treason? Are the priests seen as the guardians of the nation, or is the god supposed to be watching over it personally, or is it everyone’s responsibility? How does the hierarchy have time enough to deal with governmental duties and also attend to holy ones? What happens if a priest gets a grudge against a layman? Does he have any recourse? What do the priests teach and think about other nations outside their own, that don’t have the same god or system of government? How are ceremonies such as funerals, weddings, births, executions, and inheritance handled?
You’ve got to consider what happens to the justice system in such a society, too. The church can have an Inquisition, or interrogators dedicated to rooting out heresy, or whatever. But they can’t have just that, the same way that it wouldn’t make sense if all policemen in our world were cruel and prone to beating up criminals. People would eventually rise, and the lives of anybody who served in that branch of the church would become worth less than spit. Churches actually have an advantage here, as they can say it’s the will of the god, or have a nicer face (like the Catholic Church did) to shake its head and deplore the excesses of their iron fist. Not attending to it…well. Then I’m looking around for the rebels.

8) What is the concept of the afterlife?

Most religions will have one. Once again, yours doesn’t have to. Or it can be a fairly bleak interpretation of the afterlife, as the Greek one was. Some people got to go to the Islands of the Blest or the Elysian Fields, some were tortured for their crimes in life, but most people just spent their existence floating around in a gray limbo, reigned over by dark Hades and pale Persephone. In such a society, people might put rather more emphasis on staying alive, or at least dying in such a fashion that they would have a chance at happiness.
In a religion that has a glorious afterlife for pretty much everyone, perhaps people are less concerned about their code of conduct. It wouldn’t necessarily guarantee a society of vicious monsters (I don’t follow that idea that all morality is ultimately and only religious in origin), but it would certainly lessen the persistent fear of Hell, or demons, or darkness after death. Perhaps the people in your world even chat regularly with their dead relatives, and it’s not considered a matter of note to meet a ghost at all. That’d be an interesting place to write in.
Perhaps there are many alternatives, not just two or three. Worship this god, go to this god’s heaven. Get reborn if you want. Wander around forever as a ghost if that’s your idea of fun. Once again, it would affect the society, and perhaps people would even plan to come back as their own descendants and reap the rewards of decisions they made in this lifetime.
Or perhaps your people just don’t know, or think it’s oblivion. In that case, theories will probably be made up to fill the gap. The idea of oblivion seems to hurt and infuriate a lot more people in our world than even the uncertainty of not knowing.