Wider is bad in winter because it spreads the weight of your car out over a wider area. When you drive on snow, you DO NOT want to drive on snowshoes - in other words on top of the snow. Skinnier tires have the upper hand in compressing snow into the tread of the tire providing more grip. Whereas during the summer, weight spread over a greater area provides more friction. This extra grip allows for faster cornering, more grip during takeoff, etc. Sidewall is a completely different issue in itself. The easiest way to understand it is with the two extremes. Tires with lots of sidewall have more flex during extreme cornering which translates to a cushier ride but it affects handling negatively. Low profile tires have a stiffer ride and generally handle better. For your purposes, I would NOT recommend changing tire size. Without purchasing different wheels, changing tire size can effect your odometer/speedometer readings. Simply by choosing an all weather tire with sporty characteristics you can see a small improvement in handling.
Hope this helps!