The ‘star’ or mortice bolts top and bottom is the right solution, cheap and easy. Although fitting can be a challenge a ‘multi-point lock’ can be fitted to a timber door but talk to an ironmonger first as they are specialist bits of kit. Alternately fit an extra deadlock, 1 high and 1 low on the...
Hinge misalignment sounds likely but also the door may be binding on the stop (not the frame) forcing it back open, check for a consistent gap with a credit card when in the closed position. The other alternative is air movement (a draught), for this a concealed spring chain closer (try...
I had the same problem when my bathroom floor was tiled. Rising hinges worked a treat and they can help to close the door. Generally just a fit a pair as 3 are difficult to align
a rebate component fills the gap, they are usually 13mm or 25mm. The rebate is often listed with a decent lock, try a website doorchic.co.uk perhaps? Use a ‘mortice door bolts’ top and bottom for extra security.
I think that buying a shorter cylinder is the answer. Your current one is 100mm overall, for most 45mm doors you only need a 35/35 (70m overall). No door handles will cover this difference. You could get another shorter cylinder to the same difference
Most fire doors are 44mm but you are likely to have a 35mm door. This means that you need a new frame=messy. Since this is not a Building Regs requirment then even facing the door in ply will add to its integrity and fire resistance. Consider intumescent paint also. Smoke is the real killer so...