#1. Go to faithlife.com, and search for your group from the search box, or locate it from the GROUPS sidebar on the left.
One you are on your church group page, click on the ellipsis menu to the far right, just under your profile banner picture. Click SETTINGS. You'll then see where you can rename your church group.
#2. Interesting question that will have to be answered by Faithlife, or someone who's successfully accomplished this before. I see no options to accomplish this.
Thanks - Mission accomplished on name change
I figured out I could make one a subgroup of the other ... so I'm a step closer.