Should I glue the stopper into the wood and which glue is best?

This is a personal preference. If the stopper threads into the wood real tight, there is no need to glue. However, if the top is taken off and on often, over time the threads will get loose and it will need to be glued anyway. I use Tite Bond II which hold nicely once it dries. If you want to be able to remove the top for cleaning, I would suggest using the threaded inserts.