Since the lens caps are not airtight it simply doesn't matter at all. Put them on if you like (maybe it adds some tidiness since the caps are not "flying around") or leave them off. If you have more than one lens and you want to put them into camera bag for shooting it might come more handy to leave the caps on.
The connection between lens and body also is not airtight (only a few some models have weather sealing), you don't need to take off the lens.
Regarding RH setting: please check the manual of your dry cabinet and your camera.