hey spencer, you've taken the screws out, so i don't know why it won't come out. you might try gently popping it with a rubber mallet to see if that might persuade it. the gasket may have bonded itself to the door and door handle over the years. As to the second question, the answer is: very carefully!
if you have a truck shop manual it will give you step by step directions on how to remove the door glass. if you don't have a copy of it, go to online manuals and look there.
www.chevy.tocmp.com and scroll down to it. make sure your hands are clean and not greasy and make sure that your tongue is out and at a 45 degree angle pointing downward.
