it appeared when the sun was made
The "new moon" is technically not visible at all. The Moon is directly between the Earth and the Sun and none of the illuminated part is visible from Earth. Sometime on the day of the new moon, the Moon will appear as a very thin crescent (last crescent or first crescent).
it would still appear to rotate
The term "blue moon" refers to the second full moon in a calendar month, which happens roughly every 2.7 years. The moon does not actually appear blue in color. The phrase "once in a blue moon" has led to the misconception that the moon can appear blue at times.
The moon has two main shapes (full and new only occur once a month each, and the half moon twice, so we will disregard those for now) - crescent and gibbous. It can also be waxing or waning. A waxing crescent or waxing gibbous will appear to grow larger every night, while a waning crescent or waning gibbous will appear to shrink.
The four main phases of the moon are New Moon, First Quarter, Full Moon, and Last Quarter. These phases occur as the moon orbits Earth and changes position relative to the sun, causing different portions of the moon to appear illuminated from our perspective.
In new moon, at the end.
The "new moon" is technically not visible at all. The Moon is directly between the Earth and the Sun and none of the illuminated part is visible from Earth. Sometime on the day of the new moon, the Moon will appear as a very thin crescent (last crescent or first crescent).
It will appear as a Gibbous moon (3/4 of a full moon).
the first 15 days of the month
Mamoru (his name in the original Japanese) first appears in episode 1.
it would still appear to rotate
it would still appear to rotate
First quarter and last quarter appear as a half moon. about halfway down this page are some pictures: http://www.calculatorcat.com/moon_phases/moon_phases.phtml
because the moon and sun are so far they appear to be moving.
In August, 2011, New Moon is the night of August 28th. Note that "New Moon" visually doesn't appear - there is no moon to seeon the night of "new moon".
At the time of the "First Quarter" and "Third Quarter" phases, the moon appears half-illuminated. These occasions are roughly 7 days and 21 days after New Moon respectively, and Full Moon occurs at the time exactly midway between them. Note: Both phases are easily visible in the daytime. First Quarter rises in the east around Noon, and Third Quarter sets in the west around Noon.
A new moon may appear once or twice each month.