I have level 31 bone and now level 2 smithing (from making a bunch of these for guild points) and I still fail the synth roughly a quarter of the time. I wonder how much higher smithing has to be to make +1 ?
Crafting has nothing to do with a full moon, it does however, have to do with the days of the week (in game days). Also, some other factors include your moghancement (like moghancement: earth), and your skill in the craft (obviously).
I can make Copper Hairpins +1 every synth because they are goldsmithing lvl 11 (or so) and I'm lvl 51 in goldsmithing. Since I'm so much higher than the item, I get a lot of HQ synths (HQ=high quality).
If I make them on darksday, I get a lot more HQs, but a lot more failures as well. If I make them on lightsday, I get a lot more successes, but very few HQs. Crafting on the day of your crystal improves the odds of success, but not an HQ synth. Craft on the day that beats the element of your crystal (use earth crystals on windsday) for less chances of success but more chances of HQ synths.
EDIT: Sorry, left out fishing... which DOES have to do with the full moon (a lot more catches). But other crafts aren't affected by the moon phases.