The H is a 3.6 litre, 2H is 4 litre. The H is more susceptible to cracking heads and is a fair bit slower. About the only parts commonality between the two that I can think of is the thermostat.
Best idea would be to use the 5 speed, keep in mind that the best bellhousing to use would be an early 2h that is before the end of 1984, later bellhousings cant the gearbox over more and the transfer case hangs lower. Then you'd need the gearbox crossmember and chassis brackets from a post 1980...
The problem with your doors are the bushes/pins in the door hinges have worn, you can buy new pins and bushes for them, though you need to remove the windscreen so the top hinge pin can come out fully. Keep up the good work, she's a pretty good rig.