No FAA medical required. You FAA license will be valid as long as your JAA license and medical are valid.
Your nice little FAA credit-card license will drop through the post regardless of whether you have a BFR or not.
The requirement for a BFR is questionable. Several independent FAA offices have stated that it is NOT required for based-on licenses. That requirement is satisfied by your SEP renewal. I have it in writing that it is not required, as do several other forum members. I've been checked recently when flying on my FAA license and the inspector phoned Oklahoma to verify said paperwork and subsequently OK'd it (I did have a valid BFR sign-off, but it never got that far). I can guarantee that 100% of US facilities would require evidence of a BFR before renting you an aircraft though...