You're not crazy! Well, OK maybe you are, but I'm right there with you! :)
We had/have shower doors in what became our baby's bathroom. We removed the doors (just the doors, not the frame) and are still storing them in the garage to put back up when she's a teenager :) In the meantime we use a BabySmart Cooshee Tub Guard to cover the door frame's bottom pointy edges. It comes with a matching little pad for your knees on the hard bathroom floor, too. We got this from OneStepAhead.com. It's squishy foam and it's GREAT!
As for the germs, I would use a bleach solution around the edges, then rinse them like crazy, let it dry, then re-caulk over the ugliness. We re-caulked ourselves and it really does look brand-new. Very easy -- just a tube of waterproof caulk at Home Depot and you're done. As for the other stains, personally, I'd use whatever cleanser necessary to get it off, then RINSE, RINSE, RINSE and re-clean it after with a gentle soap to make you feel better that the harsh chemicals are "washed down the drain." You've gotta figure that whoever lived there before probably used harsh chemicals at some point. And over time you have rinsed/washed them off. So just do that for your kids in a shorter timeframe :)
Hope that helps a little, and good luck!