The fabric insert was attached to the door panels in a couple of different ways.
Either way, they're a pain to remove.
Some are just glued on.
Others have plastic pins that push though the door panel & are then melted over on the backside. Usually with some glue in there too.
Those tabs will need to have the melted over part either broken off or dremel'd off.
The glue can be separated with a putty knife.
If the insert's panel gets messed up during the removal, they are easy enough to recreate. Use the original as a pattern. The panel can be made out of 1/8" mahogany door skin or a fairly stiff plastic. 1/4" poly foam is what was used for the padding. Headliner material is perfect for that. Fabric side glued to the panel, foam side facing out.
Use 3M 'Spray 76' or 'Spray 90' to glue that on.
Don't trim the foam off square to the panel. Leave it about 1/4" bigger all around. This is will give a nice finished edge as it will wrap around when the material is wrapped onto the panel.
You can use those same 3M glues that the foam was applied with to attach the fabric.
The newly wrapped insert panel can be attached to the door panel with a good construction glue. I've used both LePage PL Premium Adhesive & Titebond Fast Set Construction Adhesive. Both come in a tube & require a caulking gun.
Don't apply it too close to the edges of the panel insert. Don't want it to seep out when pushing the 2 parts together.
Make sure that you get the panel in the right spot & aligned properly before the glue kicks. Either of those glues are going to be a permanent bond.
The carpet is a major PitA to remove. The glue GM used was meant to be forever.
Start at a corner & start soaking in something like a reducer or thinner at one of the corners. Something to soften up that factory glue. Lighter fluid (Naptha) works too, but it evaporates really fast so you can only work small areas at a time with that.
Whatever chemical you use, test it on the D/P's back side first. Some things, like gun wash, will actually melt the plastic.
Once the chemical starts to soften the glue, get in there with the putty knife. A 2" wide one works best.
Once the carpet is off, there will be bits & pieces of the that left behind on the panel. I like to go over the door panel's carpet area with a mini-angle grinder & a scotchbrite disc to clean all that up.
You can do the same on that upper area where the insert goes.
Even still, have patience removing the carpet. It's going to take a bit. GM was serious about it not coming off.