The moon part is easy(ish), the first day of the last moon means the waxing crescent moon, or scientifically; the day after the new moon, but realistically by looking at the moon it's very hard to determine (without counting days from the last full moon) exactly when the moon is new, because one day either side the crescent is so small it would be near invisible (and there certainly wouldn't be enough light from it to shine on a key hole), so it's quite likely that Durin's day would fall on the first easily visible waxing crescent moon, or two or three days after the new moon. The evidence for this is as follows; Tolkien also tells us that the sun and the moon must be visible in the sky at the same time, a one day old waxing crescent moon is so thin and close to the sun you can't spot it in the sky when the sun is also up (unless you have a modern tracking telescope anyway), the crescent needs to be at least two or three days old to be bright enough, and far enough away from the sun, to be able to see it when the sun is upthe first day of the last moon of Autumn on the threshold of Winter
The reason I never know if the date is right is because I have no idea what dates for autumn ending/winter beginning that Tolkien would have used? There are so many different definitions of autumn/fall in northern hemisphere nations from both history and the modern day, I actually think I have used a different set of dates some years than others
I've tried by best at researching season dates for Middle earth but it seems there are different ones for different regions and nations, and I can't find reference to a dwarven set of dates, I wondered if any of you know of any information regarding this? even if it's just a best guess it will be better than using the modern astronomical or meteorological definition of British autumn