Even large groups like New Here don't hurt as much as a single user with Facebook. You're fine in most cases to put a group on a shared host. Exception - when you're hosting video or other rich media locally, or you subscribe it to Facebook for easy access to Facebook page reshares. Don't do that, and you're fine.
Uhm. I've just created email@example.com, and when I try to do the very first things (change the profile photo, for example), I get a "permission denied" message. This is both if I manage it as Manolo Martínez, or if I log in as philosophy.
Do you have more than one tab/window open? Don't do that - your browser gets confused referring to invalid cookies.
If you're using Facebook/Iceweasel, you occasionally get this because of cache issues. Clearing your cache isn't enough, you need to close your browser (completely, so it's no longer around in your RAM) then reopen it.
You can also do that without having the same email and password (none of my groups share either email or password), by connecting to you group then visiting mydomain/delegate while logged in as the group.
@page promoter rotates profiles from it's members on the front page here, and above the view contacts and directory pages of this site. A small number of private servers have also included it on their sites. Since we've got some big-ish groups, lots of people see it.
New Here is really for people and meta-groups. @Let's Talk is the place for groups.
Also, make sure you're listed in the public directory. With certain configurations, this option defaults to no.
The best way to promote a group, though, is to simply post to it with @-mentions instead of wall-to-wall posts occasionally.