Here's how to disassemble and remove an engine and transmission from a Toyota Corolla.
Removing the engine from most transversely mounted, 4 cylinder cars are quite similar.
In my case, I opted to disassemble major components off of the engine before removing the engine and transmission as an assembly, including water pump, alternator, starter, AC compressor, power steering pump, air intake, throttle body, radiator, AC condenser and exhaust manifold.
This allows a lot more room in the engine bay for removal. if your doing a straight engine swap, the removal of these components is not necessary.
Once the above mentioned components are removed, there are 4 engine mounts that hold the engine/tranny to the car body. The front and rear mounts are bushings with a thru bolt. They are secured underneath the subframe by 3-4 bolts.
The driver side tranny mount is also a thru bolt. However since the tranny is being lifted up, the mount must be removed from the inner fender. The passenger side engine mount is a vertically mounted bushing, and is typically left for last.
Strapping the engine was done with a thick heavy chain, around the engine. Seat belts were used across the driver and passenger side mounts for balance.
A shop crane was used to hoist the engine and transmission up over the car, and then the car was rolled back.
6 bolts hold the engine to the transmission at the bell housing. Once separated, the torque converter is exposed. It can then be unbolted from the flywheel beneath the access plate under the engine block.
The flywheel is then unbolted, and the engine mounted to an engine stand using the engine mount bolts.
This process was demonstrated on a 2001 Toyota Corolla with a 1ZZ-FE vvti engine.