Which is correct (upon the assumption that there is only one pope at any given time):
1) He has had an audience with two popes.
2) He has had audiences with two popes.
Could '2' be used if he has had one audience with each?
Is '1' correct at all? He has met them separately. Doesn't '1' suggest that he met two popes at the same time?
Actually, there was a time when there were actually two popes. I am assuming that the sentences are about someone who is living in our time.
Your understanding is correct.