I can't seem to find anything on the restrictions and laws of a church priest or monk. I'm wondering what a monk can and cannot do do to religious bindings.

I'd say what their respective gods of worship want and don't want. Looking over at Daggerfall quests and information of the different and such may prove useful for the 8 (Talos excluded for reasons that he didn't appear till the end of DF). There's also the whole thing in MW about the cult and Temple. I doubt celibacy is a requirement for most organizations, and most likely a no no of celibacy for those that worship Arkay (life and death) and Debella (aesthetics, love, etc, etc)

what were the restriction on you in the Imperial or Almsavi Cults? Don't kill anyone without permission, don't steal without permission, don't refuse a request, don't drive over elderly people without a license.

I also assume that the same goes for Kynareth (in my opinion even more then Arkay) doing the whole "Long live nature" stuff... and that is a very important part of nature

Hey, those two godesses and Arkay are all about life :hubbahubba: . But for me, I'm way too dedicated to Khorne and Nurgle, so I prefer to either spread the wonder gifts of Nurgle or carry on the flames of war of Khorne!

Basically what the god wants. Stendarr would want you to smite evil. Kynareth would want you to be one with nature. Etc.

There are most likely several different monastic orders for each god, maybe many different ones. That gives you room to use your imagination, so I'd say you should pick restrictions and ideals that you think work best.