We've been to Germany and Austria several times in the winter and travelled both by train and rental car.
Generally:

• A car is a good idea in small towns and for local travel
• A car is not necessary and can be a hassle in larger cities
• Rental cars will have winter tires that are pretty effective in most conditions. I've never had to use chains.
• Yes, it snows in the winter, particularly in and around the Alps, which can affect driving conditions. My experience is that the Germans and Austrian are pretty good at clearing the main roads after a snow fall, but parking lots and backroads might not be cleared as quickly
• Trains work well for travel between towns/cities. We often use travel by train for day trips to another town even when we have a car, especially when the weather is bad. One of my favorite trips was from Garmisch to Mittenwald during a heavy snowfall - picture postcard views as we explored Mittenwald without having to worry about parking or driving in bad weather. Get the "Lander-ticket" for travel within a German state.