I'd like to lock a record after its marked ""inactive"" to all users except for Admin. Is it doable?

You can add to you rule that the
Inactive Oroject is unchecked.

Ie they are allowed to edit if they are the Project Manager and also if the inactive project is unchecked.

If that table of Client Contacts is a child of the Project, then just lookup that checkbox field to the child contacts and use it in a Custom Rule as well.