My idea would be to add an additional class to those fields in the field management section of CB, in order to identify them?
That's doable. Simply edit your fields in CB > Field Management and add your custom CSS class to the "CSS Class" parameter under the Global tab. You can do this for entire tabs too in CB > Tab Management.
If yes - how can I achieve this. How to deal with the group names (didn't find fields for those yet)?
There's no functionality for that, but group names already have a CSS class to denote group name. Groups are also surrounded by a CSS class matching their group id.
So for example if the grounds id is "10" then the CSS class surrounding it is "gjGroup10". Names are surrounded by "cbCanvasLayoutTitle", which is a part of our generic canvas component. To specifically target group names in the top canvas you'd use the CSS selector of ".gjGroup10 > .cbCanvasLayout .cbCanvasLayoutTitle".
Group lists are a bit different. They'll have "gjGroupsGroup10" while categories will have "gjCategoryGroup10". For those you should be able to just do ".gjCategoryGroup10 .cbCanvasLayoutContent".