Was the hymen of the Theotokos broken when she gave birth to Our Lord, or was it somehow miraculously sustained? The debates I've read on Orthodox forums about this topic are quite bizarre, and I was wondering if there was an "official" Catholic position on the matter.

Well, I don't tend to use the phrase "I'm right and you're not", but matters of religious doctrine are slightly different to arguing against scientific theory.
If someone says "Jesus was a mortal prophet" is this compatible with someone saying "Jesus was purely divine" and the Son of God? Is either compatible with "Jesus is both fully human and fully divine"