Will check the owner's manual. I think it's possible.
I downloaded a 2012 200 Owner's Manual - check page 17:
Customer Key Programming
You can program new keys to the system if you have two
valid Sentry Keys by performing the following procedure:
1. Cut the additional Sentry Key Transponder blank(s)
to match the ignition switch lock cylinder key code.
2. Insert the first valid key into the ignition switch. Turn
the ignition switch to the ON/RUN position for at least
three seconds, but no longer than 15 seconds. Then, turn
the ignition switch to the LOCK position and remove the
first key.
3. Insert the second valid key into the ignition switch.
Turn the ignition switch to the ON/RUN position within
15 seconds. After ten seconds, a chime will sound. In
addition, the Vehicle Security Light will begin to flash.
Turn the ignition switch to the LOCK position and
remove the second key.
4. Insert a blank Sentry Key into the ignition switch.
Turn the ignition switch to the ON/RUN position within
60 seconds. After 10 seconds, a single chime will sound.
In addition, the Vehicle Security Light will stop flashing.
To indicate that programming is complete, the indicator
light will turn on again for three seconds and then turn
off.
The new Sentry Key is programmed. The Remote
Keyless Entry (RKE) transmitter will also be programmed
during this procedure.
Repeat this procedure to program up to eight keys. If you
do not have a programmed Sentry Key contact your
authorized dealer for details.
NOTE: If a programmed key is lost, see your authorized
dealer to have all remaining keys erased from the systems
memory. This will prevent the lost key from starting
your vehicle. The remaining keys must then be reprogrammed.
All vehicle keys must be taken to your authorized
dealer at the time of service to be reprogrammed.
"I just learned this: You have to press the lock button first, then close the door, to set the alarm. We've been locking it after all doors closed; that apparently doesn't set the alarm."
Bob - check page 19 of your owner's manual - it should arm the system whether or not the door(s) are open with the remote. If you're using the switch on the door, yes, that door must be open. If it does not appear to do so, I'd have the dealer check it:
To Arm The System
1. Remove the keys from the ignition switch and get out
of the vehicle.
2. Lock the door using either the power door lock switch
(one door must be open) or the LOCK button on the
Remote Keyless Entry (RKE) transmitter (doors can be
open or closed), and close all doors.
NOTE: The Vehicle Security Alarm will not arm if you
lock the doors with the manual door lock plungers.
3. The Vehicle Security Light in the instrument cluster
will flash for 16 seconds. This shows that the Vehicle
Security Alarm is arming. During this period, if a door is
opened, the ignition switch is turned to ON/RUN, or the
power door locks are unlocked in any manner, the
Vehicle Security Alarm will automatically disarm.
NOTE:
• During the 16-second arming period, if a door is
opened or the ignition switch is turned to ON/RUN,
the Vehicle Security Alarm will automatically disarm.
• Once armed, the Vehicle Security Alarm disables the
unlock switch on the driver door trim panel and
passenger door trim panel, the trunk release button on
the instrument panel, and the HomeLink/Garage
Door Opener (if equipped).