About first problem, as I checked those are the Headers made by single modules.
Problem can be caused by new Joomla 6 rules and installed AMM (RegularLabs). Somehow when you are logged as Super User you see all modules, not only one assined. Maybe you need to add exclusions, i.e., specify when the module should not be displayed.
On both sites you've the same SPPB version and template, it means the same code is used. But Joomla 6 has different code compare to Joomla 5, right. That's why AMM may work differently. And we have no control over that. Maybe RegularLabs support may help. But in UE we have holidays now till Monday, anyway you can ask them.