There is very limited research on this but the observation of increased disturbance due to the phase of the moon (lunacy) is not entirely without support strange though it may seem. The effect is not just gravitational potentially but also in lighter nights and the psychological impact of observing a full moon, although why this should be important I don't know. Personally, whilst not buying into an effect I can't rule one out, it needs more evidence. The relevance apart from personal interest would be in understanding the mechanism of course. Animals also seem to be affected by lunar phases. However, there are probably higher priorities for research..........
Possibly, but I am not sure if any research has been done looking at, for example, cloud cover and its impact on the lunar cycle. Who knows what this is about! But it seems we cannot rule out some sort of impact on our psychology however mediated.