Hi Mark,
Yes, the Groups module could do with an update - I just want to release Event Templating and then I will try and work on this (unless a UserProfile update is needed first), so it is a few months off at the moment...
1) When you say Default Icon - which one do you mean? I want to evenutally change the detail screen to use Templating, so you could lay it out as you like (with the profile pic of the owner).
2) This depends on the site requirements - most social networks use the online/physical group concept - obviously not everyone does though. Can you elaborate a little more on what you require though - ie. a setting for Only Physical, Only Online or Both (or None as well maybe)?
Out of interest, some of the changes I want to put into the next Groups update can be seen on the PokerDIY Poker League module (this is a custom module that was derived from the Groups module once upon a time). You can see that you can filter on Leagues by name (first letter), and the leagues have images that are shown in the poker league search results.