Having recently removed/reinstalled the entire engine and having removed only the the transmission (twice) in the past, my $.02 is that it is easier to just remove the trans alone. Not by much, but there's a lot more connections to bung up by removing the whole engine. Engine left in doesn't help you much with alternator access, but after removing the front crossmember and both drive axles to get the trans out, you should have decent access to the alternator.
Tips?
1) Things are easier to remove than work around. When in doubt just unbolt something. Fair game is anything to the right of the engine that could get in the way (header tank, intake, transmission mount/damper, etc)
2) Label all bolts/nuts that come out of the car and hoses/wires that are disconencted. There aren't too many things that are involved here, but masking tape is your friend.
3) Depending on where the car is located (lift or front on jack stands) may determine how to get the trans out.
Front jacked-up (how I did it) - Have a number of jack stands to strategically place in order to jack the engine up. I had to move mine once or twice as I jacked up things and removed engine mounts. You'll only have the left side engine mount to attach it to the car. You'll want some sort of engine lift to lower the trans to the floor, then slide it out from under the car.
On lift - Not sure how you'll support the right side of the engine, but I'm sure it can be worked out. Borrow/rent a transmission lift to lower it out.
It's really not that bad if you take your time. Ask away on any questions you have - we'll be more than happy to help out.
Rumors of my demise have been greatly exaggerated.
"Track Elan" Monaco White Elan / "Peely" Calypso Red Elan / 2007 Subaru Legacy GT-Cobb Tuned / 1993 Honda CBR600F2