The fuel quantity gauge on my wife's 2000 Mustang is permanently stuck at 1/2 full. How do I check the system to determine whether it is the gauge, or the sending unit in the tank?

It's most likely the sending unit

easiest way is to drop the tank and check the sender with a volt meter. You're going to be looking for resistance as you move the level arm up and down. If it maintains a constant resistance, then the sender is bad.

If you show a reading at the sender itself, then check at the back of the cluster for the signal. If signal is going into the cluster, then it would be the cluster that needs to be fixed.

If the cluster is the problem, I recommend getting a used cluster, and "splitting" it by using your circuit board, but the replacement front section. This will prevent you from having to pay the dealer to reprogram the keys and remotes.