You probably can't hurt anything by putting a little additional torque on the head bolts, but by the same token if it is having a problem under the proper torque more torque is very unlikely to cure anything. Obviously do not go and throw a half turn on a bolt - you do not want to break a head bolt!
When I have pulled the engine I leave the transmission. Lots of people recommend pulling them together. It's a matter of preference. For me a handful of bolts in weird places is less of a headache than wrangling the powertrain out at a 45 degree angle, but go with what suits you.

I remove the radiator because it's stupid easy and it mitigates risk, but it's not even remotely necessary. Do lay a piece of thick cardboard over it to prevent damage if you leave it. Any decent cherry picker will get an engine onto a pickup, no problem. You might consider replacing pistons, but assuming there is no damage it's certainly not necessary. Same with the rods. Crank is pretty burly, nothing to worry about there. Do replace the oil pump.
The OTC 5609 is $60 and is a very adequate tester. I can't imagine buying regulators and gauges will save that much, but I have no actual idea.

Just a thought.