Darrin De May
I'm Awesome
Not sure about the latch as I never really paid attention to it when I had mine out, but the door can be removed very easily. You just open it, then push in a little on the plastic sides and pull down if I remember correctly.
The glove box on my 2000 C3500 is a little different. I had to take it out when I swapped out the original glovebox light bulb with an LED bulb. The right side of the glovebox releases by pushing in on it as described, but the left side is held on with a plastic cord with a hook on the end that attaches to the side of the glovebox. There's a release tab on the hook that you lift from outside of the glovebox, or push on with a screwdriver from the inside, then the hook slides out easily.
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Right after I swapped out the glovebox light bulb and put the glovebox back in, I went to swap the right side underdash light bulb, and it broke while I was attempting to extract it, so I pulled the glovebox again to see how to remove the socket. Turns out I didn't need to, as I was able to grip the filaments on the broken bulb with a hemostat and pull it out. Hemostats are handy for purposes other than being used as a roachclip!