IANAL, but I have to say no to deal with questions like this all the time, so here's my understanding. (But ask a real lawyer if you need a real opinion!)
Can a college theatre department sell (at a prophet) videos of past productions of non-original material-- that is, the production is riginal and we paid for rights to put it on, but the playwright is still alive?
When you paid for the performance rights, there was a contract (which may have just been the form that you filled out and signed) that specified just what you could and couldn't do: I'd be amazed if your right to make money off the play extended to anything beyond the original run. I expect that (as an educational institution) you could make a good case for showing them to future classes and the like, but not for making money from them.
It seems to me no, but that if we are just asking for fees for materials to make copies for the cast or something, then it is OK?
Legally, still probably not okay, but practically speaking, I can't imagine anyone enforcing it if you do it for the actual cost of a tape. Making money is another matter, and I really doubt they'd care if it was being sold to a cast member if they decided they wanted to be dickish about it.
How about on-line streaming of entire productions (not exerpts for advertising purposes)?
Not without the copyright holder's permission. Again, that's something that's likely to be addressed when you bought the rights.