I would never walk a dog alongside the bike. I did in an earlier decade till my dog looked back only to see a dog pack behind it, and then proceeded to run right under my bike. If your dog will fit in a dog trailer, that's what I would vote for.
You probably want to get yourself a utility bike. It could be an old MTB, modified for the street with slick tires, fenders, a rack and lights. Or and old 10 speed or even a comfort bike. Or you could get fancier and add an Xtracycle rear end. You want a bike that will be serviceable in all weather, and one that is not likely to attract the attention of thieves.
You also need to get some winter clothing. Check the Winter Riding forum for good ideas. Seems like everyone needs something different.
You can do it. Good luck.