When transmission is shifted into reverse the car will not Move

You most likely have an issue with a seal inside the transmission or a solenoid issue in the valve body. Unfortunately there are no easy ways to figure this out other than to use a scan tool to check for codes and then you will need to monitor the solenoid applications to find out if one is failing to apply in reverse. The fact that it works when it is hot points to a mechanical issue with the solenoid itself. Meaning the control module is applying the solenoid but it can't physically operate. Then when it gets hot it is able to operate properly.

Let's start with checking for codes and then we can go from there. Thanks