There are many things to cause your engine to seize. The most common are :-
a motor can not run with out oil, the motor will seize up and can break parts
Engine, not motor, and, while not impossible, such a swap would cause a lot of problems, not the least of which would be the car engine not being protected against saltwater corrosion.
check motor mounts
Not really. Without oil, the engine will soon have metal rubbing on metal, which will cause parts to go hot and seize. And that means engine destruction and expensive repairs.
No you shouldn't your oil level may be fine, but your oil pressure may not be. Would hate to seize the motor.
it will seize up
The motor which blows air through the vents.
Your motor oil going bad would cause that. That is in the crankcase, not the transmission.
NO
A loose engine mount would not normally cause the engine to fall out because each car has several motor mounts. A loose mount could cause the motor to be off of balance.
The engine will overheat and eventually seize. This will pretty much destroy the engine if it is driven long enough.
If you run the car without oil ..you have more problems than the injector rails.. you cant actaly damage them with lack of oil .. however ..you will cause the motor to overheat and seize ..you will also stuff he pistons and the block ..the crank shaft will be hot welded to the piston rods..the valves at the top of the motor will be stuck and they will be worn down ..running a motor without oil ..would result in a whole engine rebuild ..or the cheaper option of replace the whole lot ...DONT RUN THE CAR WITHOUT OIL ..