You may be able to find a nice portable shelther for less on sale at northern tools or harbor freight. How much snow do you get during the winter? If not too much, you could easily knock it off the shelter to avoid problems stated above. If you get 10 inches of snow in one day, maybe one of those aluminum or other metal roofed car ports would be better. Letting a project sit out in the elements will create more work for you than you want. Just look at all those nice rust free old cars from the dessert, vs. the cancer ridden carcasses of 6 yr old cars living outside in New York. Weather and salt are hard on steel.
It's a tough one for sure. I remember while living in Wyoming, during the winter, that my room mate and I would get up around the same time and try to start our cars. When it was really cold, we jump one off the other. Then I'd plow a path out to the main road in my old landcruiser. Snow was so high one day I heard a bamm! Got out and found somebody had left their car in the middle of the road, under 3 ft of snow. Didn't phase the landcruiser, car was gone when I got home that night, think the snow plows might of hit it too!