Yes, heat melts snow
could be the electronics (garage job) or it could be the alternator (but unlikely because its a new car)
Yes, you can use overdrive in your car when driving in the snow. Using overdrive helps the car maintain better traction in slick and icy conditions.
Yes and used to be common place to have a set in the boot of ones car about 10-20 years ago when it snowed often in UK winters, although local police will not take kindly to you accidentally leaving them on when the snow has cleared as they can cause damage to the road surface.
Because the car is not moving
cause they do
Yes, either can cause a shake while driving.
yes
a bad engine
yes
Yes, it is possible.
Yes, it can.
Bad tires and bad alignment can cause the car to shake at certain speeds.
Possibly
A clogged muffler can.
Yes, when idling and driving.
yes