I hope they tested the anti-freeze/coolant. Even -2C is still freezing. However, a cold start by itself (with no other issues) wouldn't blow a head gasket.
Starting and running the engine for short periods has been know to cause a lot of white smoke on subsequent restarts. Usually from condensation or combustion byproduct that the engine/exhaust didn't get hot enough to evaporate. So the next start-up you see the cloud. However, that wouldn't throw any codes or cause "check engine" issues.