You didn't explicity say you wanted 4wd but for this purpose I'll assume you do. That being said I'd alway reconmend winter tires - $WD may help get you going but it doesn't help you stop.
There aren't any subcompact cars with 4wd in the US. In the compact space a Subaru Impreza would be a good option. The XV crosstrek is simular but tends to sell for a little more money.
The latest batch of compact CUV like the 500x, HRV, CX-3 are above you price range but there optinos in this size with models that have been out a while longer and from brands that many would disregard
Mitusbhi Outlander Sport in in your price range. A 2013 CPO, LS 4wd is in your price range and has the balance of the 10 year powertrain warranty
Suzuki SX4 might be a perfect fit - 2013 was the last model and these can be had for under $10,000 for a well spec'd model.Reliability looks good for the older models
Nissan Juke is another option but personally the styling of it is not to my tastes.