Yes the screwdriver slots are to turn the shut off valves.
On some valves, putting the slots horizontal makes them fully open.
Others rotate a full turn or more, like a hose faucet handle.
Trial and error, unless someone here knows best.
I would guess horizontal.
Yes, the pressure balance cartridge is in that chamber between the main body and the cold shut off.
Applying some penetrating oil on the threads for a day might help it loosen up.
I don’t see any screws holding the valve onto any framing, so if you try to remove the nut to get to the pressure balance cartridge, you should put counter pressure somewhere else safe on the valve, with another wrench.
Don’t try to grab the main round part you are fooling with now, that part can twist and get ruined or pinched.
BTW, I just saw on another forum where someone squirted a good dose of straight vinegar into each hole where the springs came out of, with a turkey baster, then quickly packed the whole thing with paper towels soaked in vinegar overnight, and their flow got much better.
CLR would be much stronger, but not sure if safe to expose to the brass surfaces for a long time.
So your shutoffs could be clogged, or not in the right position, or the little chambers leading to the main cartridge could be clogged, or the pressure balance cartridge could be shot, or any combo of these.