The older cars get, the less reliable they get.
For safe handling and braking in the winter, I'd set aside $800 or so for a second set of wheels (steel is fine) with good winter tires.
My personal preference is for compacts. They might not be as powerful as a midsize with a V6, but they're much more agile and fun in typical daily driving.
For reliability, newer is always better. One value buy I often recommend is the 2008-2011 Ford Focus in SES (sport) trim. In this trim (denoted by dark multi-spoke alloy wheels) the suspension tuning is European. Not the most stylish cars, but they don't look bad in blue.
The Acura TSX recommended above is a very nice car, but you might have to get an old one with many miles to stay within your price limit. Same with a Subaru Impreza. Subarus tend to be reliable early on, but can get expensive to maintain once over 160,000 km.