Alex,
Invites:
The new Invite module is coming in 4.1. It will be a vast improvement for tracking invitations, custom ones, and a lot of other good stuff. Still finishing up some things now, but 4.1 will be out this month. It's going to utilize the Requests system from Facebook, so there will be a jewel notification in the top right when an invitation comes in. PM's are a bit more difficult for things that are meant to be 'broad, predefined' messages, like invitations are. One nice thing that we're expecting to include (still in progress though) is the ability for admins to send requests to users that have connected, ex: "Check out the new blog post". I'm sure there will be issues and other concerns, so the old JFBCInvite module will be included, for now, but will be phased out.
The new requests module has no invitation limits. We're investigating filtering out already connected users, but aren't sure that will be in 4.1 initially.
Fanbox:
Like the idea. Hadn't thought about that.. but something like you mention is something we've done (haphazzardly) in the past. Right now, the Invite module only shows to users that have connected.. when it could show to any Facebook user. Why just connected users? Who knows. I'll definitely take this down as a task to investigate an option for "Show to: Connected Only, not connected, or all" on all social widgets (that it makes sense on).
Thanks for the suggestions, and hope that helps,
Alex