I did some work on this today. I used a 100 grit 3" sanding disc on my right-angle die grinder to cut/sand the stakes off of the speakers, then test-fitted the new speakers. Then I decided to just cut the posts off completely since they weren't helping anything. I cut the screw head off of some #8 screws, drilled holes in the black plastic backing on the trim panel, and used JB Weld to epoxy the studs into place. Once this cures I'll mount the speakers and see how it goes. I drilled the holes slightly undersized, put a drop of JB Weld in the hole, and threaded the studs into the holes. I then added a bit more epoxy around the stud. The studs are machine threads so they're not cutting much of a thread in the plastic, but it's enough to hold it in place while the epoxy sets.
When I took the passenger side trim piece out, the top of the black backing plastic was broken off, so that top clip and its surrounding plastic was missing. I'm the second owner of my car ('06 bought in '09) so I wonder if I'm not the first person to contemplate speaker replacement in my car...
2006 Solstice, Mysterious 5-speed