I don't know about the Fit specifically, but I drive a Yaris (Toyota equivalent). I live in a winter snowy place in the intermountain west, though it doesn't stay snowy the entire winter (freeze-melt cycles). Good quality snow tires are the key. I drive fine in the snow. (I experience some hills, but not super steep ones, and remember these cars are light, so YMMV.) Also not sure about clearance, but years ago I drove a Mazda 626 and actually took it on forest service roads (lol). Pretty low clearance on that car, and I actually bottomed out a few times (but survived without damage). If you are careful, and the roads are not actual two tracks, but just dirt that are occasionally graded, it's probably fine. (But I would take a look and judge for yourself.)
(Subaru's are fine cars too, my sister drives an Outback and my parents drive a Forrester. But AWD is not cheap on gas.)
I say buy the Fit!