Doors – Is the interior door LH or RH

Your plan to fix the gap in the center of the door sounds fine. I might suggest using some shingle shims instead of flat stock however. If you want to just remove the interior trim only, that is fine also. Using the shingle shims, place the thickest end in first and by trial and error, adjust the opposing shim to a size that will snug in nicely and then by gently tapping it in next to the first shim, remove the excess gap. Put a small finish nail through the jamb and shims to prevent them from moving. Drilling a pilot hole will assure not splitting the shims with the nail in inserted. Now just trim off the excess and reinstall your trim. Job done! Good Luck

I see in your picture that there is a wooden shim between the door frame and the jack stud. Typically the frame should be screwed into the jack stud separately from the door hinge. The small door hinge screws should only have to hold onto the frame of the door and usually don't need to be driven into the wall stud.

A fire door can be quite heavy though and might need a bit more.

You can see here that they make door hinge hardware that comes with at least one extra long wood screw that will anchor your door hinge to both the door frame and the jack stud. This will also further strengthen the door frame itself.

Go to the door accessory section of your local hardware store and find where they keep plastic packs of door hinge hardware. Each hinge typically comes with short screws and one long screw. Each pack is cheap, usually under $4, you will need three. Make sure you match the shape and hole pattern of the existing door hinge.