I think others have covered it, but just to summarize the key points:
* Yes, when the moon is close to being new, that means that it’s in roughly the same direction from us as the Sun, and so will be rising close to sunrise and setting close to sunset.
* Yes, the the time between one new moon and the next is about 29.5 days. So, yes, if this month’s new moon occurred around 6am (i.e., around sunrise/moonrise), you can expect next month’s new moon to occur around 6pm (i.e., around sunset/moonset).
* No, there’s no reason to expect the moon to be rising at the instant that it becomes new.
* (In fact, the moon’s position relative to the horizon depends on where you’re standing. If it became new at 6am in New York, then it will have become new at 11am in London, because time zones. So if New Yorkers saw it become new while rising, Londoners will have seen it become new while high in the sky.)
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