What make the steering wheel shake is the tires that need wheel balance.
Yes A VERY common concern is steering wheel shake due to ice in the rims
Yes, it can
Normally this is caused by one or more of the tires out of balance. If the steering wheel is shaking then you know it is a front tire. Have all your tires rotated and balanced.
the steering wheel wont make the front wheels make noise as there is no generating source from the steering wheel. it will be the other way around.
bad bushings on the rack mount making it loose hence turning stering wheel will make it loose and bounce == Have the rack/box pipes been tampered with? Is it manual or power steering? Does this happen whilst driving or if the vehicle is stationery? www.powerbrakeandsteering.co.za
jiggle the steering wheel there is probably tension on it to make it hard to unlock
Check the fluid in the power steering. If it is low the car will make noise when you turn the wheel.
Power steering or wheel bearing?
check your power steering fluid