1) I don't think it is even possible. The clamping mechanism that holds the chip to the motherboard can't be lifted without removing the cooler first.
2) Bad idea anyway. Best to clean and reapply paste.
3) If your copy of windows is a retail or upgrade copy, you are legal to move it to a new PC(aka motherboard).
If your copy is OEM, it must legally stay with the motherboard. If the motherboard is defective and needs to be replaced, and an exact duplicate is not available, microsoft is understanding. They will usually authorize the move so long as you assure them that it is being used on only this one pc.