my window is the same way. There is a little plastic piece that goes around a metal ball on the regulator, and the plastic piece goes inside of the window rail, alows for up and down motion while giving a little side to side motion as well. Just like mine, the plastic piece probably broke. Sadly the whole regulator is like 200 bucks from GM, you cant just get the piece.
To take off the door panel. Simply pull out every bolt and clip in the door, then there should be pop rivets along the bottom and the side of the door. If the panel doesn't pop off, you missed one. I've taken mine off soooo many times, and I have the same gen as you.