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.
its old or something wrong with the car
Could be in need of a tune up.
No
what chater
wheels need balancing, sometimes shake is only evident at high speed
what shake it up character would definitly perfer to travel by car not airplane
yes
check motor mounts
Engine mounts
Normally this is caused by warped brake rotors.
possibly a broken or missing engine mount