1: If the case is bad, you will get wheel hop when turning sharply at slow speed, such as a parking lot. You may also hear noises from under the truck like groaning or a scraping.
2: what can it damage? Depends on just how bad that transfercase is (or more directly the viscous coupler within it).
It can bring a quicker death to upper/lower balljoints, tierods, will kill CV's and differentials and drain your wallet quickly if it is ignored, (dont ask how I know).
3: The jury is out on this one, actually. I say no, but I removed my front driveshaft long after my transfercase was dead. My Tcase was so dead that the viscous coupler was frozen up solid and the truck never moved when in park for the year that I drove without the front shaft installed. I say that you can not kill what is already dead, and you may just save other parts from getting harmed if your transfercase is dead now.
How do you tell? Take that driveshaft out, go park the truck on a steep hill, but don't get out of it. If the truck rolls slowly down the hill, your viscous coupler is good (at least it is at this time).
Heres a question for you: What makes you think your AWD case is either dead or damaged?