1a) The absorbent material will have been partly used and so the nappy will feel firmer. Staying in a damp nappy for any extended length of time can result in a rash, but in my own experience, it's likely to be mild, perhaps slightly itchy. In the mornng it's important to wash, obviously, and maybe open a window just in case of pee-smell. I guess it's down to personal preference, but I find sleeping in a damp nappy as comforting as a dry one, personally.
1b) I have only done this once, just because I usually put one on for bed having been to the toilet so don't have a lot during the night. The time I did though, I got quite a rash. The nappy felt tighter and quite comforting. By the morning, the nappy will have absorbed more deeply or dried slightly, it feels different. I think there's something comforting in waking up in the morning, needing a change.
A note about leakage:
As any time of day or night, if you max out a nappy then sit or lie on it, it can squash some pee back out. Also, if you use it whilst lying down, think about how the position you are in will direct the pee. Usually, lying on your back or front (or turning over to use both) is best because it uses the full padding capacity. On your side, the pee runs into the leg cuffs and is more likely to leak.
2) I haven't tried to achieve this. I'm not sure it would be a good idea. I'd imagine though that the first thing is becoming able to relax with using the nappy in bed, being confident that it will not leak, or using bed mats so leakage isn't a worry. Once you can use the nappy without straining, but by just 'letting go', I guess that's a first step. If you are fighting against your own muscles, stop, relax for a bit and try just letting go in a few minutes. Don't risk harming yourself.