I was backing out of a parking spot in the mall and the car behind me wanted my spot. There was also a car in front of me eyeing my soon to be relinquished spot. I started to back out of my spot and then pulled forward. I felt I did not have enough clearance to avoid hitting the car in front of me, so I put the car in reverse again to back up a little more. In the process of me putting the car in reverse and turing around to look back and gauge the distance to the car behind me, I felt an impact. The question that I have is this. Is there a required amount of room that other vehicles are supposed to give the vacating vehicle before attempting to secure the parking spot?

No, but a driver is required to exercise reasonable care.

No, but the driver pulling out, if they don't have room, should not move and allow the other cars to move on or leave.

If they were stopped and not moving, you have to either not move if you can not get out safetly. So if you were moving and hit them, and they were parked, you are at fault even if they were stopped to a point of not giving you room