Is it possible to create a field group that can be modified across entry types?
No, field groups are just a way to have your fields grouped in the field layout designer and fields index page. Another advantage is the possibility to drag a whole group in the designer to create a new tab.

Approaches, other than a max 1 Matrix field you mentioned, that I can think of are:


multiple predefined rows in a Table field
multiple predefined rows in a Set Table field, which allows you to have a read only "Label" column and set a max rows
custom solution implementing a new field type that has multiple input fields