Are you sure that the cars main 12v battery under the bonnet is healthy? Modern cars are heavily reliant on a good 12v source of power to keep all the electronics happy, and when a battery is failing all sorts of weird electrical gremlins start to appear.

This cold weather also kills any batteries that were already getting weak - they dont like the cold weather.

As for stopping the alarm going off in the middle of the night - the simple solution is to park it safely in a garage and leave it unlocked.....but this might not be an option for you.