Can one get an STC w/o a patent?
I'm not aware of any necessary link. The main problem would be exposure to litigation if one used a patented or registered design gadget without an appropriate licencing arrangement.
If so, would FAA have to develop its own specs for certifying each Approach to the add?
The applicant (ie the organisation applying for the STC) needs to show compliance with the Design Standards so the effort lies with the applicant rather than the regulator. The regulator's task is to ensure such compliance to minimise the risk of dodgy things getting into the mainstream .. experimental category is for doing one's own thing.