UPDATE:

I went to the local auto parts store and found the key fob RF/battery testing machine. Pressing all buttons lit up the tester so the key fob is functional & battery is good.

What I'm not completely sure about is, if the car starts and operates fine when the key fob is inserted in the dash, does that mean that it's propertly coded to the car's CAS. Or does locking/unlocking and starting/running the vehicle different functions of the programming?

How can the key start and run the vehicle but not be able lock/unlock the alarm (if the diversity antenna is functional).