cracking the engine over is all battery effort and then the alternator looks after the electricals from then on while also putting charge back into the battery. 5 mins wont be long enough to put sufficient charge back into the battery for a full charge. plus 5 mins of winter driving means extra fuel to try and bring the car up to temperature quickly.
i would say the starting issue will have been slight flooding and lack of charge in the battery also. the warning lights matter, again from lack of charge in the battery. few weeks ago while driving with a failing alternator the air bag light came on, purley down to the battery having a hard time.
ecu reset or clear the codes with a reader, long drive and you should be all good.
david