The aircon certainly won't run with the engine stopped, because the aircon pump is run from the engine rotation. The heated rear window and mirrors will still work, and the heater fan motor will run, but will blow cold air because the water pump will not be pumping hot water into the heater matrix (water pump only turns when engine is running). I hope you didn't have your headlights and wipers on as well?
For your battery to go down so quickly (5 minutes) I am assuming that it was in a low state of charge anyway, short runs and local miles in the winter are not enough to keep the battery well charged, you need to do a decent long run (20 to 30 miles) at least once a week. If a car battery is continually kept in a low charge state (below 50%) it will die from sulphation, when the plates develop a white coating, the batteries do not usually recover from this and will not charge fully ever again.
Your battery needs to supply over 200 amps to start a cold engine, that's why its good policy to buy the biggest A/h battery that will fit your car.