It's still going to be a maintenance hassle. More stuff to break, tires issues, more stuff getting in the way when repairing something else. etc. Plus there is still a MPG and performance hit from having the extra weight.
No. Possibly in a couple of counties in the westernmost parts of MD, but the rest of the state no. Not enough to cause rust issues even if you keep a car for 25 years and never do anything rust preventative, or even wash it in the winter.
Is there another outside cat in your yard? I've seen plenty of episodes of My Cat From Hell where the presence of an outdoor cat near your house caused an indoor cat to become territorial. Did the vet check for a urinary tract infection?