First, you need a utility knife and a putty knife to cut away the caulking. Under the caulking, you will fing glazier's points. Little triangle shaped keepers that hold the glass in the space. A flat screwdriver and a small hammer can be used to renove those. The piece of glass will then lift out in one piece, which is MUCH safer! Now, if you break the glass, you will still have to do all the above. It will be much more dangerous with pieces of broken glass to work around.

If it has a proper protective finish on it, it shouldn't "stain" at all. Things should just wipe off with no damage as per the cabinet line passing the KCMA testing. Now grout and mortar have a strong chemical bond to just about anything. Removing it means abrasives and damage to the underlying surface results from using abrasives, including scotchbrite. Replacement door time for that, paid for by the one who damaged it.

Believe it or not I had a quartz counter get a long line on it from a pineapple! It was on the counter and when I moved it to a new spot I guess I slid it and it left a mark that wouldn’t budge no matter what I tried (including acetone). I finally put some blue dawn on a magic eraser and than goodness it worked (magic eraser alone didn’t help at all, nor did dawn alone, nor soft scrub etc).

Vinegar, WD-40 or Alcohol will make it easier to remove but you always need to use a razor blade and just scrape it. Try to get the most of it off with the blade and then use a paper towel with some vinegar to rub off the tiny fragments. A vacuum helps to remove the tiny particles.