Get a piece of wood large enough to cover the hole, attach that to the wall, then attach the thermostat to the wood. Since the edges will be visible, use solid wood, not plywood. 1/2" thick should be plenty strong. Quick, easy, cheap, and you won't have to deal with patching the drywall.
My cheap solution for small patchups, is to use a 6 or 7 inch piece of a paint stirrer stick.
Smear it with wood glue or construction adhesive and center it on your hole, inside the drywall, aimed where you might want to drive your screws. A screw in the middle can be a handle.
You'll have to keep it place while it dries. If you don't care about more holes,...
Think the screw has nothing to do with removing the dead front. Think this is how:
Turn the switch to OFF.
Lift up the top plastic section using the holes on the side. This will allow lifting or prying the bottom section over the stop on the bottom.
But it may be that you just pry on the bottom in the central slot.
Highly resinous pine studs can get very hard with age. You should drill the pilot hole the full depth that the screw will go into the stud then a little past that.
Often the diameter of the pilot hole must be increased over the recommended. The drill you used (1/16") is very small. When you get your new screws you will probably need to use a 1/8" ...
I was able to extract the broken screw from the deck board by backing out all the "good" screws in it and flipping it over. I discovered that the screw broke before it got into the joist so I carved out a hole around it (in the side that faces down) with my Dremel until I was able to grip it with my locking pliers and unscrew it so ...
For the screw that is stuck... The easiest way to deal with it is to chop it off and then to take a bigger bit and drill it down a good 1/8"+ so it doesn't ever stick out. Sure you can spend a ton of time backing it out but I wouldn't bother with it.
I will give you two general pieces of advice when doing something like a deck...
Test out your screws ...
Finally removed the screw from my Duravit toilet as suggested by @RedGrittyBrick (kudos to him) using the back of a claw hammer, a rolled toilet roll cardboard cilinder as fulcrum and to avoid damaging the porcelain and quite a lot of brute force.
Those are "Mollys": (hollow wall anchors)
They don't come out,
the screw comes out (it's long so it comes out slowly), but the anchor part will stay.
It might be possible to drill out the anchor part from the front
Or maybe cut off the arms at the back, remove the nut fold up the stubs and eject the ...
One non-glue option that comes to mind is a threaded insert, specifically one suited for MDF. I did a quick search for "threaded insert for MDF" which returned many. One promising link is EZ-LOK Soft Wood Inserts as it references MDF among other types of wood.
Two versions are noted in the linked page, providing a metal foundation for the removable ...