I don't think your battery should be empty from the cold. About the smoke and badly running engine, if theres no major problems it should clear up after the engine is properly warm. The smoke from the radiator is possibly steam from an exhaust heating up, it could be coolant, I'd keep an eye on it. Your hot mechanic neighbour sounds great, put a piccy up. (That applies only if you're male or lesbian and the neighbour is female ) Honda's are pretty much bulletproof, riding an old Kawasaki in the winter is much more of an adventure. Keep safe.