Was there a time when a Pope had a beard or moustache, or both ? And at what time in history?

Answer 1

Yes.

Peter as a Jew would have had a beard: http://www.gsp-ind.com/popes/about.htm

With love in Christ.

Answer 2

Loads of bearded popes, especially those who were members of bearded religious orders.

Oddly enough, I can only think of one bearded pope in a picture off the top of my head and that was Julius II (1503 - 1513). He wasn't a nice chappy either. Watch the 'Agony and the Ecstasy" but disappointingly, he had no beard in the film.