Languages are in continuous evolution, due to geographical proximity, conquests, trade, etc. In Romania/Moldavia the Fanariot-Greek/Ottoman rule left many words in the Romanian vocabulary (which is 70% based on latin) of which:
καρπούζι- harbuz = watermelon, and another similarly sounding
πεπόνι - pepene = melon
are still in use today.
I've only ever heard "kharbazeh" refer to the honeydew; also not sure about dialecticality, but I've only ever heard the pronunciation "kharbāzeh," with which the dictionary disagrees. Modern Persian uses "hendevāneh" to refer to watermelon. I'm not sure we have a word referring to melons as a generic class.
It should be a /u/. I, as a native speaker, did not really understand the difference, so both are ok. Ου is not a diphthong either way... I believe that it sounds a bit different due to a glissando she does and the way she colors her voice, but it's barely noticable, at least to me.