phpBB is designed to do most of the requirements that you wrote. Also, invisionboard fulfils all your requirements.
On both boards:
disallow anyone from registering. You should be able to use the admin side to add new users.
change the settings on the forums so that you have to be logged in to do anything
I'm pretty sure that both the boards I mentioned above have the facility to make sure that you (the admin) have to verify each post before it goes live.
A note about number 3: if you only have a few users that won't be posting very regularly, then this is fine. However, if there are going to be a lot of posts made, it might be worth getting someone to help you out with this (maybe make them a moderator?)
Hope this helps.