Delayed starting

NONE OF THIS HAPPENS WHEN THE CAR IS FULLY WARMED UP.
    The remote does not work to unlock or lock; it does when warm.  Otherwise, door locks work fine.
    If the key is turned immediately to the "START" position, the starter hesitates one to two seconds before engaging.
    The clock goes to "12:00" as soon as the key is turned to the "ON" position.  (The radio pre-sets remain.)
    When the engine is started, the indicator on the rear defrost switch flashes for about a minute.
    Turn the ignition off, it hesitates briefly before the engine stops.  
    When opening the driver's door, with the key OUT of the ignition, the alarm sounds as if the lights are on or the key is still in (until the door is closed).
    The alarm will sound ONCE when re-entering.