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.

Long underwear is great for being warm without wearing a jacket or bulky sweaters all the time. I live in DC, and, last winter, my basement bedroom was unheated. Long underwear + lots of blankets meant I didn't care too much. Flannel sheets may also be your friend.

Please, let me know winter life hacks/survival tips/things a Texan would not know about cold weather.

The first thing to keep in mind is that winter weather is quite variable. Where I live (MA), the temperature (in Fahrenheit) in the winter ranges from around -10 to +50. It's good to have ways to keep yourself warm at various points in that range.

Going down human anatomy, I would recommend:

Head

• A warm hat that covers your ears
• If you are into outdoor exercise, a headband to keep your ears warm; they can get very cold even if you're producing enough heat to keep the rest of your body warm
• Sunglasses; snow can cause a lot of glare. Make sure they have full UV protection (most do).
• Optional: scarf and/or ski mask; on the very cold days, especially if it's also windy, it can be helpful to keep your face covered.
• Chapstick; the air is pretty dry in the winter.

Torso:

• A moderately warm jacket and a very warm jacket, for varying temperatures; I use a modular ski jacket, which has an outer shell and an inner lining. The lining can be easily detached and used as a stand-alone jacket on less cold days.
• Sweater: depending on the temperature at work, you may feel you need one. Also, at home, wearing a sweater is a lot cheaper than cranking the heat up high.

Hands: * Get a good pair of gloves. I cannot emphasize this enough. Preferably get gloves with some grip to them, to make things like opening doors easier. * Moisturizing lotion; as mentioned before, winter air is very dry.

Legs:

• Most people don't really do much differently here.
• Optional: Long underwear can be helpful, depending on how easily you get cold.
• If you will be getting exercise outdoors and are male, wear an extra pair of underwear, ideally not very breathable. Why is left as an exercise for the reader.

Feet:

• Insulated boots are very useful, though not absolutely necessary. In urban areas, there are often a lot of puddles from melting snow, so make sure they're very waterproof, too.
• In a less urban area or on a large college campus where you may have to trudge through deep-ish snow, I would also recommend gators to keep the snow from getting inside your shoes.
• Snowshoes and spikes are unnecessary.
• Wool socks are helpful, depending on how easily you get cold.

Other things:

• I find a sun lamp helpful; YMMV.
• People spend more time indoors with less ventilation, so diseases spread more easily. Take precautions as appropriate.

My mom broke her ankle when she slipped on some ice that was covered with snow (so that it looked safe to step on). 99.9% of the time this is not an issue though (and the snow is safe to step on...

Having been depressed before and having recently read "The Mood Cure" by Julia Ross, I think I'm going to be ordering a lamp soon.

Do I need snow shoes?

Unlikely. However, depending on average snowfall and where you walk, you'll may want waterproof boots that come up to at least your ankle (Gortex or equivalent). This also depends on how much time you plan to spend outside.

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

Depends entirely on how much time you're outside. I tended to walk everywhere in my college days, so I wore an insulated coat. You'll probably want gloves, even if you stay inside your car most of the time. If you're outside a lot, consider a knit hat, especially if there's a lot of wind where you live.

Other things:

• Thick socks are awesome for wearing around your house.
• I like having really cheap fingerless knit gloves for typing inside.
• If you do spend time outside, look into flannel lined jeans or khakis. Thermal underwear works too.
• Heating can cost quite a bit of money; set your thermostat lower in the evenings and load up on blankets.
• At home, drafts are your enemy. If you're in a newer place, I'd assume this shouldn't be as much of a problem. If you're in a place with single pane windows, think about doing putting up window insulation film. In college, this helped with heating costs, but they have the downside of clouding the view.
• If you have your own driveway, you need a snow-shovel. Even if you theoretically have someone paid to shovel the snow.

If you don't care about walking on ice, get a nice pair of some lined duck boots (Sorbel, LL Bean). Expensive but totally worth it I think.

I recently found out about the disposable hand and foot warmers. Not perfect but they do a decent job for being so inexpensive (around $1-2).