Welcome to Gaia! ::

Yes, there's celebrations for the full moon, since the moon is the basis for the months. It's called an Esbat and some folks bake cookies that look like full moons. The most powerful are those that fall closest to the the cross-quarter days and the fire festivals. I've heard of some celebrating on the full moon rather than the calender day. New moons tend to be a time of rest, though some celebrate the second new moon in a calender month since it's sometimes called a fairy moon.

That depends on what type of Pagan you're talking about. Its not one religion, but rather a category of religions. That's allot of options to cover with just one word. Yeah, some pagans might celebrate a full moon and others might not. It all depends on what pagan religion you're talking about. That is not weird at all. Many people are captivated by the moon. That's why most primitive societies thought of the moon as a God or Goddess (among other reasons). I donno about "moon festivals" or anything, but some traditions will have a ceremony to welcome the new moon. Of course, that's the new moon, not the full moon.
I odn't put much stock in zodiac signs. Its more likely your rules by your belief in your sign rather than the sign itself. Also, moonstones have nothing to do with the actual moon. But, yes, it does seem just a bit impulsive.