I think this depends entirely on the religion and denomination, so it's very hard to give a broad answer.
I'll answer for mine:
1) Yes. His job--not just what he's paid for, but his mandate, spiritually speaking,--is to do this. He's not just a teacher or mentor. He has a duty to confront the members of his church when they do something our religion says is wrong. He does this out of love and concern for the members' spiritual well-being, not just because he's in the mood to be nosy and intrusive or tyrannical.
2) Our church--many churches do, actually--has a very well-defined approach for this: confront in person, privately, then bring in someone else (not just any random person, of course; it's not a gossip session), then to the rest of people in the church who are in authority to take disciplinary action. I think that in the case of people who are still under their parents' authority (minors living at home, in other words), he would have the right to make them part of that process. I think the alternative--to have them find out only once the final step of disciplinary action was taken--would be far worse.
3) This is a tricky one. There's the case of people who are trying to change behavior but working against very strong desires and temptations. I think in that case, he should be discreet about it for a time and attempt to help them change. I think he should lean on them heavily to tell their parents, as they're also the spiritual leaders of their families (assuming the parents are part of the same church). Then there are cases where they have no intention of changing behavior, in which case, in our church, our pastor would have a duty to continue the process.
4) I would absolutely want to know at some point, but I'd rather hear it from my own kid. I would definitely NOT want to find out only when disciplinary action had been taken.
All that said, when you become a member of our church, you know that the pastor and elders do have these responsibilities. If church discipline takes place, it's no surprise, as it's always at the end of this process. It rarely comes to that anyway. People either decide they don't subscribe to those beliefs anymore and leave on their own, or they stop what they're doing.