Will putting your bed on the floor help with back pain issues?

Results may vary. It probably won't hurt to try, but success isn't assured. Firming up the bed probably decreases the curve your spine is asked to conform to while you sleep, which may decrease pain. If this worked for everyone, there wouldn't be dozens of other remedies out there.
No. No, but laying down on carpetted floor without a pillow and bending knees for 4 hours will help most back pain.