what happens if I delete a group member who is already logged in games? Will that ruin anything? Can I move gaming group members between groups?

There are no issues with de-activating a Player with logged games. They will just not be available to pick as a Player for new played games, but any games they've played in will be untouched.

In regards to moving Players to different gaming groups, there currently is no ability to do that. We have an email exchange going on. Once I understand the feature better I'll put an idea up on here.