While creating/viewing/editing an issue, can I control which fields are visible and to which group?
You can select which fields are shown while creating, editing or viewing an issue by changing the Screen(s).
In Gemini 4 this can be done via the Administration > Screens menu.. Please note that you have to assign the screen to the item types by editing each item type (via the admin section).
In Gemini 5 this is is also done through the screens menu but the navigation is trhough the Customize -> Process menu. Please see this document for more detail on how to do this.