Sounds like wheel imbalance or possibly a bad wheel bearing(s).
Yes, either can cause a shake while driving.
Nope.
Loose motor mounts.
Maybe its your motor mounts.
Yes, when idling and driving.
Most likely it has unbalanced or bad tire(s). Bent wheel rims also can be the cause.
only in the morning? cold morning, warm morning? how many miles do you get in you travels before this happens?
Have your tires checked out! Body dithering on the driving comfort and handling judgment will have an effect. In the case of a transmission shaft deformation caused by the unbalance, generally, this happens in the chassis of the car collision more, but a lot of auto parts have shockproof.
Wheel balance may be off...
car starts but shut off while driving
Nobody could answer that question without looking at your car. One of the possibilities is that your car is overheating. Check the coolant. By the way, if this is the case, and you are driving your car without coolant, stop driving right now because your car is in danger of being irreversabely ruined forever.
If it only happens when you are braking I would suspect warped brake rotors. Other reasons the front end could shake would include unbalanced tires or the vehicle needing realignment. Whatever the case may be; driving a car in this condition is unsafe and you should repair it or have it repaired before you drive it again.