Old Gray Mare:
for my 2 cents-
Either I missed it or you never mentioned if the snow on the roof had melted. I would agree that quite possibly the snow melted, and what spring was left in the roof, let it rise up enough for you to back the tractor out. As for the tires...I also believe, unless you have seen or noticed 'a brine leak' on the tire, that the tire is fine. Even with the brine loading, you must still keep them inflated to required pressure. As previously stated, move tractor so that valve stem is at the very top, and fill tire to recommended pressure. Keep an eye on it, and if no brine is seen leaking, you should be okay. As you have stated, nearest tractor dealer is 75 miles away....where is the nearest tractor tire shop? They are the ones with the expertise, equipment, etc to repair/replace a bad tractor tire.
Also, living in the north east, we experience temperature changes all the time in tire pressures. It is very common. We went from a really quite warm 'fall' to a relatively cold 'winter' just recently. My wife's recently installed snow tires all had to be inflated again to pressure due the drastic temperature drop....I am talking 8 pounds per tire. The same holds true for summer tires installed in early spring....as temps warm up, inflation pressure has to be dropped or they will be overinflated. I am not sure about your location in Arizona, but you must notice a difference on you pickup truck tires from winter when temps are in the 30-60 range and summer when temps hover in the upper 80's to over 100.