I don't think it's as much of a concern in today's very nearly sealed fuel systems as it used to be in the days when they were vented on both ends, but I suppose that it could get stale. Probably takes a long time (many months).
There are many other things that could cause a car to seem to run better, including gasoline more appropriate to the weather (it's pretty complex stuff and they tweak the formulation on a time scale of weeks), seasonal differences in the use of oxygenators such as alcohol or MTBE, different brand of gas than you bought last time, and beneficial effects of steady driving at highway speeds in its own right after a winter of chugging around town.
As for effects on the gas pump of running the tank low, I suspect that (as long as it isn't so low the pump is cavitating or anything like that) the main risk is a big dose of whatever ook was in the bottom of the tank. This is said to be especially bad for late-model GM products, under the theory that they have marginal fuel pumps anyway.
I used to run my old T-bird intentionally low just before a planned change of fuel filter when I was first resurrecting it and some very fine rust would come up from the tank or the lines. It cleaned up nicely over time in this manner, and whatever's left down there, if anything, I'm happy enough to leave alone. NOTE: I mention this just as an example of how you can have stuff in the nether regions of a fuel tank. That's a much different design than anything on the road today, and I certainly do not recommend doing this to anything that has a pump upstream of the first real, changeable filter, which covers pretty much anything modern that has a pump embedded in the tank.
Cheers,
--Joe