Let's see, I'll try to answer to your questions...
1. I do indeed think that the Moon has been summoned by the power of Majora, by a few logical and theorical reasons: first off, in the Moon, as you pointed out, we can see a big face. Second off, rarely does the Moon show itself before it is night, and I think it is impossible, if not almost impossible, to see a Moon appearing in daytime three days in a row. Then we also have the theorical part, by which the Moon is a creation of Majora, and the exterior part you mentioned shows some of Majora's crazyness.
2. Majora is a monster sealed in a mask, probably a very ancient monster, so i think it must have dozens, hundreds or even thousands of years. In this adventure, Majora Mask's takes the possesion of a kid, that is, Skull Kid.
3. The five kids playing around a big tree inside the Moon are not random, and there are a lot of theories trying to explain why the kids are in there, and many of them can be perfectly applied to these four kids. In my personal opinion, four of these kids, the ones with the masks of Odolwa, Goht, Gyorg and Twinmold represent the four giants which are friends of Skull Kid, and the child with the mask of Majora represents the Skull Kid, probably possesed by the mask, who, thinking link is a villain, gives him the Fierce Deity Mask so that Link represents a villain when fighting with Majora. I also think these kids might be related with the words 'moonchild' and 'lunatic'. Moonchilds and lunatics are people who think they are from another planets, considered as crazy, and these five kids might be moonchilds, lunatics, not only because they literally are moonchilds, but because they might have their minds as crazy as Majora's. The trees they are dancing around might also represent something, as trees are many tiems used to represent things as life. I'll end here so this post won't be as long as the personal theory I have.
I hope I pretty much satisfied you with my answers