bama13time

Hello
I have a 04 GX470 electrical issue. My wife had to get jump started yesterday at work so I sent her down to the local battery store to get another battery. (I was at work) So she gets there and the guy installs the battery and says the alternator is not charging. He said this is more than likely why the battery went out to start with. So she drives it home about 5 miles and parks it. So I get home and start to change out the alternator, which is a total pain. So I get the new alternator in and crank it back up and I notice right off the bat the engine is running rough. So I take my volt meter and check across the battery and with the engine at idle is only putting out 13.4 and starts dropping all the way to below 12v. I shut the engine off. So that’s were I am at. New battery and new alternator but still not charging battery.

I just called Autozone to have them check out the other Alternator and they said it was charging?

So questions.
1. Is he something else in the wiring harness to the alternator
2. Could I have damaged the new battery on the way home from a 5 mile drive.
3. Is the electrical issue making the car idle rough. (I think it is)
4. The old alternator was ok and something else was wrong (like wiring harness )
5. Any ideas on checks?
Thanks for any help in advance.