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.
Some clouds may appear to be behind the moon due to the relative positions of the clouds, the moon, and the observer. When clouds are at different altitudes in the atmosphere, lower clouds can appear to be behind the moon, especially when the moon is low on the horizon. Additionally, atmospheric perspective and the moon's brightness can create the illusion of depth, making it seem as if clouds are positioned behind the moon when they are actually further away.
In new moon, at the end.
It will appear as a Gibbous moon (3/4 of a full moon).
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).
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".
A new moon may appear once or twice each month.
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.