When I think 'snow car' I automatically thin of Subaru for some reason.... I guess that's probably exactly what they want!
These Imprezas start at $17.5 and have standard Symmetrical All Wheel Drive....great for the snow.
I don't have an Impreza but I've been thinking very seriously about making it my next vehicle purchase, because I think it's an excellent vehicle.
I completely agree with you about Winter and Subaru some how being synonymous. I used to own an Outback and that was the only car I felt comfortable driving in the snow. I went through the mountains, passes, etc. and never had any issues. Subaru's drive really well, handle and perform great, and are very stable, great on gas, and durable. I would highly recommend anything from Subaru.
Check out the Toyota Matrix. It starts at $18,845 and has AWD as an optional upgrade. This might put you a bit over 20g's, but it's one hell of a car. My neighbor drives one and it's pretty slick.
Maybe you could look for a slightly used one, that might put you under budget.
Subaru would also be a great choice.
I would have to recommend the Suzuki SX4.
It's great in the snow - All Wheel Drive. It's also well within your price range.
It's also small and gas efficient. definitely a good option to consider.