Glad you got it sort of running!
I'd agree that it could be the alternator, its wiring or its ground. It grounds through the bracket to the block, so clean that up too while you're in there.
It does seem to be doing something though. No connection of the alternator would have your battery draining from 12.7v down. Below 12v and it's dead.

What is the battery at now with the car off?
You should charge the battery with an overnight charger now, instead of trying to do it with the replacement alternator.