I agree about the black tank.
So far as the grey tank, if open when it is below freezing, ice will accumulate in the sewer hose and may eventually block it or restrict flow. A hot shower may melt it, but might not. So, I'd close the grey tank valve when it is cold and leave the black tank closed until it is time to flush the tank.
I can't see how mice get into the system unless they are living in the sewers. That does happen in large collector sewer mains where there are dry places, but in the pipes in the campground, I think it no more likely than mice coming into your house from neighborhood sewers. Sewer gas should not be a problem so long as there is a water barrier in the P traps and the toilet seal works.
The thing I've wondered about is whether a mostly empty tank or mostly full tank is easier for the furnace to keep warm. Water will store a lot of heat compared to air (I think the number is 1,000 times more), but also takes longer to warm up a full tank. Keeping the grey tank closed probably keeps it warmer because of the amount of hot water that goes into it. The fresh water tank is filled with colder water in the winter from most city water systems, so maybe it is best to fill it in the morning so it has time to warm during the day for free if it is above freezing. Seems like the uric acid in the black tank won't freeze very easily. Adding salt to the black and grey tanks sometimes is suggested, but I would think it may corrode seals.
I think there is also a limit when temps get close to 0˚ F. At that point, the furnace may not keep the tanks from freezing. And when driving, some water in each tank will slosh around and prevent some freezing, or leave the furnace on as some people do. We've driven with the furnace off when the temps were around 15˚ without a problem, but if the temp had been that low all day, we might have had problems.
I also agree about RV salesmen.