A good place to start is to check for leaks in the vacuum lines associated with the heater controls. Many of the controls are vacuum actuated & if there is a leak in a control or associated line it may cause a rough idle
Yes, it can.
what chater
No
its old or something wrong with the car
Could be in need of a tune up.
what shake it up character would definitly perfer to travel by car not airplane
wheels need balancing, sometimes shake is only evident at high speed
yes
Yes it will.
possibly a broken or missing engine mount
check motor mounts
ice in the fuel system