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No, it will drift around beam on (side on) to the wind and waves until repaired. A cruise ship will roll in this case and can be uncomfortable. Gladly it doesn't happen very often.
Sounds like cruising's not for you.
Most unlikely, it's not like an aeroplane which has to keep moving. At least you won't starv.
Most ships have more than one engine - most of the bigger ships have at least four (of which they are using two at any one time while the others can be repaired). There are secondary / back-up power sources should the ship lose power so this wouldnt be a problem.
They do dock and the engine is off then, no they do not sink in port either.