Low speed problems like that are usually either a bad tire, bent wheel or a problem with CV joints.
What make the steering wheel shake is the tires that need wheel balance.
Not the steering wheel shake itself, however what is causing the steering wheel shake, can certainly be extremely dangerous and potentially cause you to lose control of the vehicle. **GET IT CHECKED OUT BEFORE THAT HAPPENS**
Yes, it is possible. Any loose/worn front end part can cause steering wheel to shake.
Yes A VERY common concern is steering wheel shake due to ice in the rims
usually your cars out of alinment
Your power steering pump and or steering box may be about to give out on you.
There are several causes of this but the most common we call a head shake. This is when the tires shake side to side, the cure is to insure the steering is tight and you add one or two hydraulic steering dampers.
Wheel balance
im guessing it has something to do with velocity and road conditions. yet again, if i'd imagine that a steering wheel shouldn't shake.... that can't be good.
Loose or worn steering parts, suspension worn, tires out of balance.
try the steering stabilzer
Sounds like warped brake rotors .