When any engine has a rough idle in park or neutral, it usually indicates that the firing order of the sparkplug wires may not be correct. A new engine usually means all new parts. Bring it to a shop and ask them to make sure the firing order is correct.
Make sure they connected all the engine mounts and the transmission mount as well.
You engine is probably misfiring. Check your ignition and fuel systems.
If your car shakes while underway, your tires have worn incorrectly from misalignment. If it shakes while in park or not moving, your engine is misfiring.
Timing belt was installed incorrect and engine is out of time.
a bad engine
No, not unless it is so low the engine is overheating.
possibly a broken or missing engine mount
because it is going on a rollercoaster of car sensations.mmmmmmmmmmmm.
this car is known to have an appearance of a rough idle even when everything is fine and a history of short-lived motor mounts
Bad engine cushions can make the engine shake. bad spark plugs cause malfunction as the car starts running.
The energy being released in the engine is not transferred to the outside; it manifests as vibration, which is transferred to the rest of the vehicle, hence the shaking of the car.
the engine mounts could be bad
Well, voltage wise, you have a lot going on when your are cranking over the engine, so some flutter, or shake is normal.