How do I remove trim around the rear window?

Yup. It's a lot of work, but it will be worth it in the end. The clips hold the moldings & the screws hold the clips to the body. Clean up the screws as best you can & let them soak a day in penetrating oil, I swear by PB Blaster, make sure the heads of the screws are as cleaned out as you can, use compressed air in the heads if you have it, so you get a good grip with your screwdriver, they are #2 screw heads, I think. (Use the correct screwdriver bit) You will be able to remove more of them that way. You can always drill new holes if you have to, & screw the clips in there, (their position is not set in stone left to right, as long as they stay in the center up & down, on the horizontal pieces at least) next to the old screw holes
Looks like the clips and screws are holding it in. They are so rusted I have to drill each one. Ugh