My grip strength is good (powerlifting) but I've learned to just let go and regrip is far easier and more annoying!
However it does take its toll on your fingers, I tend to wrap them in tape which does help.
Yes bruises and friction burns.
In most situations "floor is death" is probably correct!
We do striking a couple days a week, so it gives you a good blend of floor work and protecting from strikes whilst trying to get into dominant position on the floor. However, I find also you can get control of people without necessarily going onto the floor, although to score points in a comp you do need to either pass the guard, or get them on their back or of course submit them.
Another thing my club does is wrestling, and that has helped my grappling skills, especially in no gi where there's a different technique than relying on grips.
It's amazing how you can be in a compromised position on your back, but actually get the sweep, transition and get position on someone fairly quickly. Being on your back is never comfortable when you know a flurry of blows is likely to come!