Are there hotels within a reasonable walking distance from the main train station? Suggestions?
Dozens and dozens of them. The RS book describes many of them and shows them on a map. We stayed at the Hotel Basilea which is about 4 blocks from the train station. We've also stayed at the Hilton Metropole which is across the river and not close to the train station but they have a shuttle that leaves every 30 minutes or so and goes to the train station.
We stayed at Hotel Bretegna which is about a ten minute walk from the train station and on the Arno River near Ponte Trinita. It's walking distance to almost every site. It's in a historic building with an elevator and they have wide price range which you can check out at their website or on booking.com. We were there for five days in March 2009 and enjoyed our stay.
The old city center is very walkable from the train station. We stayed a block from the Duomo and had no problems walking to our hotel with our luggage (we traveled with 1 bag each).