All it means is don't shoot anyone or get in any gun related trouble while you are off post. As my old Drill Sergeant used to say, "If you're going to do something wrong, do it right."
Absolutely, at least regarding Alaska. You'd want to brief your people about local rules wherever they are - those are much more restrictive in Maryland, for example, than Texas. In Idaho you can be arrested for owning too few firearms... ;-)
I agree with you but not this policy. I'd rather be judged by 12 than carried by 6.