as a dog groomer i do short cuts all year round. As long as they live indoors then its fine as when they're outside they're running around to keep warm anyway and when inside I presume you have the heating on! When they get older I suggest that they either keep their own coat or have a coat to...
if they're only getting an hour a day that is most of your problem, young excitable pups should be getting at least 2 hours a day and you will find every aspect of their training easier because you wont have to fight through the frustration and excitement to get their attention!