What kind of glue do I use between the windshield and the surrounding rubber?

Butyl or a good urethane winshield sealant. Try Napa or 3m

No, to do it right you cant just glue it. Any glass company is going to recommend you remove the glass and re-install it. Thats to cover their liability, and with good reason. You could use a silicone to seal it, but it wont be right unless you have it done by a good glass company (and they are right - they probably will break the old one on the way out) Your insurance company may cover this under glass coverage.