I'm in the Midwest (so some, but not huge amounts of snow lately) and have a kid and love my '08 Prius.
The Prius handles well in the snow due to its weight (although like any vehicle would recommend snow tires which make a huge difference). The only issue snow-wise is that the ground clearance is pretty low so large drifts that an SUV could get through would hang up a Prius, but I haven't had an issue.
There's plenty of room in back for a carseat - I got the car when my daughter was 3 and she's now 6 and there's plenty of room for her even with the large Recaro brand seat we use. In fact, my husband, my in-laws, and my daughter (so 5 of us, 1 in a car seat) have taken the Prius on multiple 12 hr drives to visit family - so yes, there is a decent amount of room in the backseat.