After running first aid on the Mac HD and the Mac HD Data, should I run first aid on the MacOS base system?

It is actually best to run First Aid on the hidden Container. Within Disk Utility click "View" and select "Show All Devices" so that the hidden Container appears on the left pane of Disk Utility.

Also even if First Aid reports everything as "Ok" click "Show Details" and manually scroll back through the report looking for any unfixed errors. If you find any unfixed errors listed you will need to erase the whole physical drive and restore from a backup or clone. Disk Utility is a terrible app as it lies to its users and is unable to fix Apple's own file system issues.

You can run First Aid on the physical drive if you want as it will check the partition table.