Instructors are nearly always PIC when teaching
A tricky one. I know what you mean, but what about instrument training (in VMC)? Does an Instructor have to be PIC then? whether he wants to be or not is of course a different question.
Interestingly, most examiners I've encountered state specifically that they are
NOT PIC.
AFAIK in the US, there is no (legal) restriction to flying right seat, unless the a/c POH or placards state otherwise.