Hello, yes, the moon on the page is supposed to reflect what the moon is like outside (northern hempishere anyway - it's "upside down" down south). At the moment there's only a small amount of the moon illuminated as it's only two days till "new moon" (ie no moon visible). Once we get past that the moon will start to reappear "from the right".
The phases are
new moon
waxing crescent
first quarter
waxing gibbous
full moon
waning gibbous
third quarter
waning crescent
Crescent/gibbous just mean that less than / more than 50% respectively of the visible "disc" is illuminated. Waxing and waning just mean growing and shrinking of course.
As to why we have the moon at all, I believe it's a nod to a combo of the old idea of women being generally influenced by the moon, combined with the folk wisdom that everyone goes a bit mad on full moon.
Am dying to rephrase the previous paragraph to incorporate the phrase "ladies troubles", but fear it may be taken seriously and lead to mocking.....