The moon has lots of symbolism in its phases as stated above even stands for hidden knowledge and hidden aspects of the personality. The moonis also the symbolism of the astral gates and of water which you can gain protection from the element. Even the element of emotions reason why the actual moonstone's metaphysical benefits "soothes emotional instability" and to cause emotional harm if you put it into curse workings with magick. Also as they say most crimes are commented on full moons hint it effects the zodiac sign and planetary correspondences on that day.
That depends, I agree with Agiel's response. It's not mandatory to use this energy but it can help; it is said that the full moon is the period that the Moon is closest to the Earth and so the energy levels are strong. This is why it is said it adds strength to the spell.
It's not only that though, I sense Right Now (like, truly Now) that moon is associated with secrets... It can revel hidden traits of personalities lile Kingu said, I have learnt this on first hand during a full moon.
I believe there are a lot of things and symbolism behind the moon; what I've mentioned is not everything about him/her (I mean the Moon), you have more to learn! ^_^