Do our subcompanies in the partner system need their own account or can we manage it on our Admin company (=supercompany)?

This is up to you. You can connect as many companies to your DocuSign Account as you like. So the subcompanies can create and connect their own accounts or you as a Supercompany provide them an account.

Please be informed that there is no synchronization in place for DocuSign integration! Furthermore, supercompanies do not have access to the Settings page for DocuSign integration. DocuSign integration is handled solely on company respectively on subcompany level.