If we set up a forum on a UBB board on our server, can we move it (all threads) to another UBB board on another server later? A group wants us to host a forum for them to help them decide if their members will use it.

Also, can we set up two separate UBB boards under our URL and on our server in case the group wants us to host a board for them.

And finally, if we do that, can we move the board (all forums and threads intact) to their server one day?

Yes to all. You'll just need to own one license for each active installation.