Yes.
Yes, bad spark pugs or plug wires can cause the engine cylinder to misfire which will cause the engine to vibrate.
Yes.
as a result of bad spark plugs
Contaminated fuel could cause a misfire.
yes
the motor mount is bad
Yes.
yes
Yes.
Yes they could be cleaned and battery charged
Yes, either can cause a shake while driving.
Yes. Totally! If the engine mounts are bad the whole car will vibrate and run bad. Get it fixed.