What is the specific term for time of the night when there is no moon? There are times when the sun goes down and it takes a while for the moon to rise, because the moon obviously doesn't rise with the sunset whole year. I guess this happens only when the moon is full. Also, in some days, moon sets in the midnight and there is no moon on the sky for the rest of the night. So, I was wondering what that time period of the night is called when the moon has gone down, or has yet to rise? Can we call it 'moondown'? I am not talking about the new moon when there is no moon the whole night. Please help, and thank you in advance!
You are right that the moon only rises as the sun sets when it is full. At other times there is some time each night when both the moon and the sun are both set.

The terms moonrise and moonset are well established, by analogy with sunrise and sunset. The time when the moon is at its highest point in the sky is known as lunar culmination. You can use "lunar day" for the 24.84 hour period between lunar culminations. I know of no specific term for the period when the moon is set, or for when it is risen, instead you can say "after moonrise" or "before moonset", as appropriate.