In the long term, keeping the tires covered is not a bad thing. It will definately help on the sunny side. Covering them for a weekend is a little obsessive, though, but doesn't hurt.
Another benefit of good tire covers is, it will keep the lug nuts and brakes from getting rusty. This can happen from just weather and dew cycling, but parking near sprinklers and lawnmower blast really accelerates the corrosion. But not during a camping trip.
For long term storage, supporting the trailer by the frame is also good for the tires, not to mention the axles. You have to balance the benefits (hard to quantify, really) against the risk of potential damage due to jack slippage, which can get real expensive.
It seems I love the mountains and deserts more than my friends do. I sure miss them!
1971 Streamline Imperial project "Silver Snausage", 1985 Coleman tent trailer, 1964 Little Dipper, 1975 Northwest "Proto Toyhauler", 2004 Harbor Freight folding, still seeking my Airstream.