Reading the moon for a dark night
What matters is not the phase alone but whether the moon is up while the sky is dark. A full moon is up all night. A first-quarter moon sets around midnight, leaving the early morning dark; a last-quarter moon rises around midnight, leaving the evening dark.
The green nights in the calendar are the ones where the moon stays out of most of the dark hours at your place. Pair them with the forecast for clouds and the Milky Way's core.
Questions
Which moon phase is best for stargazing?
The few nights around new moon. A thin crescent sets soon after the sun or rises just before dawn, so most of the night is dark. From first quarter to last quarter the moon is up for much of the night and washes out the Milky Way.
Why do moonrise times change so much from night to night?
The moon moves about 13 degrees east against the stars each day, so it rises roughly 50 minutes later each night, more or less depending on the season and your latitude.
Does a moon below the horizon still brighten the sky?
Very little. Once the moon has set, the sky is essentially as dark as on a moonless night. That's why a waning moon that rises after midnight still leaves a good evening for the Milky Way.