I've been fortunate to have never had to keep my TT in a storage facility.
But, first, regardless, storing at home or storing at a facility, your questions apply to both.
Second, there is no right or wrong way to do it. Do what works best for you and your circumstances, and your attention to detail.
Third, the harder the surface you can park your camper on, the better it is. Not so much because one surface is better for the tires, that's been debated to death. But because of ease to get the camper rolling again, and what happens to the surface directly under the tires.
Parking on dirt only, the tires are subject to sinking in the mud when it rains. Hooking up and pulling out of it's "hole" may be challenging, causing your tow vehicle's tires to start spinning and the camper never moves.
Parking on board is always a good thing, regardless if on any surface. Tires will not sink, but the boards may sink some, especially in soft dirt, mud, rain.
Asphalt parking may cause the tires to form dimples in the asphalt if it gets very hot.
Concrete parking is good, and probably best. But usually concrete parking spots are usually very flat, and water could pool around the tires. Although the tires are rubber, they are still subject to the elements.
So, in my opinion, whatever surface you are storing on, put something under the tires.
About stabilizer's. I always put my down, all of them. I live in a very high wind area, and I want as much surface touch the ground as possible. I did have one camper move on me in the wind. If my jacks would have been down, it probably would have never happened. Jacks down will provide more support, and also help keep a bit of load off the tires. And if anyone attempts to steal the camper, they have to take the extra time to raise the jacks, otherwise they'll end up tearing up something under the camper dragging on the road ... that's attention they do not want.
If jacks are electric, I don't think it matters where they are set. If they are hydraulic, then you probably want them in the best position where there is the least pressure on the hydraulic fluid.