You are correct --
Once you prove a man served in Civil War, and produced a National Archives copy of his service file, that's all that's necessary.

The question about why he Givens was dropped from the rolls was a matter of curiousity, not eligibility.

I'm not sure that a Mobile City Directory exists for those years. To check, call Southern Collection - Birmingham Public Library at (205) 226-3665 and ask the librarian what years are available. The librarian can be asked to make a copy of a page and send it to you.
If you're curious, here's the main reading room in the Southern Collection --
http://www.facebook.com/pages/Southern-History-Department-Birmingham-Public-Library-System/144834798910274#!/photo.php?fbid=359166114143807&set=a.318672924859793.73775.144834798910274&type=1&theater