driving slowly

When there's snow or ice on the roads, there's really no speed slow enough that you can count on never losing traction. After the first heavy snow, you might want to practice in a low-traffic area until you get the hang of recovering from a slide. Also practice driving as if there's a full glass of water on your dashboard that you don't want to spill.

Do I need snow shoes? Spikes?

Nobody uses those for day-to-day walking, but you might want a pair of insulated boots depending on how much time you plan to spend outside. These are pretty convenient if you want to walk on ice.

Can I expect to safely walk on the sidewalk in the winter without slipping and hurting my everything?

As long as you're careful.

Do I need more/better coats than what I have now? (A duster and leather jacket, both fairly thick.)

Maybe. Again, it depends on how much you'll be outside. You'll probably want gloves.

Will I need to get a sun lamp/UV lights for the middle of the winter when the sun is up for a shorter time?

I guess, if you want to give yourself an artificial tan at home. Or are you talking about light therapy for depression? Those aren't designed to emit UV.