I have done lots of engine swaps over the years. The trick is to balance the engine on an angle to allow an easy match-up on the bellhousing. I never remove the trans with the engine -way too much work.
Use the innermost two intake manifold studs and nuts to attach a loop of lifting chain. I use two lifting eyes from the exhaust side - borrow one from another Volvo owner or car if you can. The engine is then supported at its center, and you can twist and move it easily. The balancer rigs are more trouble than they are worth. With one hand on the waterpump and a balanced engine, you can line it up and push it home.
As mentioned by another poster, leave the mounts on the engine, and remove the nuts from below the crossmember.
I always remove the hood - customers don't like dints or stressed hinges.
Take the harness out with the engine, and leave everything on the engine except the AC compressor.
If it's an automatic, remove the converter bolts. I made a tool for this, but a deep offset 17mm box end will work well. DO NOT remove the converter attached to the flex plate. Grease the converter snout well when re-installing. Be sure the converter spins easily on the crank before installing the bolts again, otherwise the converter support bearing, pump, and seal will be damaged.
It's a lot of work, but it can go smoothly and without frustration if you are careful.