The problem would most likely be that some oil got onto some of the engine. I believe there is a special cleaner for that. I have an understanding, that oil on the engine can cause some damage. ??? Oil on the engine will burn and smell/smoke...but won't cause shaking. Driving without the oil cap can lead to all sorts of junk getting into the valvetrain. If enough debris gets in there...you can score/scratch cam lobes, interfere with rocker arms, valves, etc. Excess moisture can lead to water mixing with oil which is never good. Change your oil and inspect the condition of the used oil. Look for water or excess contaminants. Also, open your radiator cap and check your coolant. If it looks milky brown (oil in the coolant looks like coffee) then you have a blown head gasket...definitely worth the time to check.
oil chane
Yes, either can cause a shake while driving.
wheel balance
Yes, when idling and driving.
You bet it can. A bent or out of balance drive shaft will cause serious shaking. A defective U-Joint will also cause it to shake and is very dangerous.
I don't know you tell me.
Tire out of balance or bent wheel.
It might be a broken motor mount or transmission mount.
It needs an front end alinement or your tires & wheels need rotated
Sounds like a misfire caused by electric supply to plugs.
bent wheels or a broken cord in a tire or a deformity in a tire.
check u joints on driveshaft.