most likely the radio not turning off when the doors are opened is due to a failed driver's door ajar switch, located in the doorlatch, and prior to replacement you can try to spray some PB Blaster into the doorlatch liberally and slam the door a few times to attempt to free the stuck switch, if that proves to no avail the switch will need to be replaced. As for the keyless entry not operating properly, first check fuse numbers 2, 16 and 22 underhood ,as well as num ber 20 inside fusebox, if fuses are good, attempt to reprogram the remote fobs using the procedure below. If reprogramming is unsuccasfull then the Ford scantool will be needed to monitor inputs to the module as well as to check for any DTC's and likely you will need to have the dealer, or a shop with the IDS scantool diagnose the concern
NOTE: All remote keyless entry (RKE) transmitters must be programmed at the same time.
NOTE: Programming (or reprogramming) of the RKE transmitters can be accomplished using a diagnostic tool or by carrying out the following steps: