How to calculate when the moon is highest in the sky to the earthbound observer? Like when is the new moon highest in the sky? Or third quarter? How do I know this for any general moon phase? Thanks.
The Moon moves round the ecliptic. At full Moon it is opposite the Sun. Hence when the Sun is furthest South (in the Northern Winter), that full Moon is that which will be furthest North in that year and so on. But plainly, the Moon moves round the ecliptic every 28 days, so every 28 days or so it is furthest North etc.

In other words there is no answer based on phase or time of year alone.