You can do most of that through use of the CP. However, if it is gonna be a public group and the leaders not able to use the cp or Mod options, then the group leader itself has to take care of making most of that through use of his/her skills.
The groups I commanded one time had prestige and awards and all that stuff. We all had our own group logo and lobbies, where only members of that particular group were allowed to go in and post, the others could not. This, as you say, Cynthia, was great for group meetings. In there we would discuss strategy for a certain event, we would discuss who was gonna do what and we would discuss alot of stuff in there.
The activities I ran there were the following:
Art Contest, Writing Contest, Mug Contest, Sprite Contest, group Art Creation, group story contest, siggy weekly contest. Aside from that, we had games, arcades, quizzes and other stuff. It was so active that I found my hands full of stuff to do as an Administrator. But I'll tell you this, I made the group thing work like hell. :D It wast the best damn forum I ever made, and I had like 50 members. the site was called The Game Club, but I deleted it once some member jacked control of it and made a mess.
One thing is for certain, though. When you have a good guy who knows what he/she is doing, things go well.