I personally do not. One reason is that I want to keep the shafts on the jacks clean and corrosion free. Also, I don't want to give intruders (mice, insects, etc) another roadway into my coach.
If you do store with the jacks down, my manual recommends lubricating the rams weekly. Every 2-3 days if in a salty environment.
As for keeping the tires healthy, I just try to drive it at least every two weeks. So far I've been able to keep that schedule for various reasons. I don't know if I'll be able to sustain that for the long run though...
The in-laws keep their tires covered. I haven't gotten there yet because most of the time I have other rigs parked around me in storage, so my tires are shaded most of the time anyway. I might start covering them though. And I'll probably start treating them with Aerospace 303 just to keep some of the UV off them.
I also keep the fridge doors wide open, and after I install a vent cover, I'll start keeping the vent open as well.
-2018 Greyhawk 29MV
-2015 Jeep Grand Cherokee Limited (Primary Toad)
-1994 Jeep Wrangler YJ (Secondary Toad)
-2014 Jay Flight 28BHBE & Ram 2500 6.4L CC 4x4 (sold)