Q, though powerful and quite funny, isn't a god. Gods can't be outwitted, and yet both Picard and Janeway have defeated him on these grounds several times, but I digress.
Anyway, I hope it's not too late to add my two credits to this. As GrayLensman has said, the systems must work together for the series to make any sense. Therefore, I have come up with a theory that I believe accommodates both systems. It contains some ideas that have already been presented, but there are some modifications.
I tend to lean more towards the CT elements forming the basis for the universe, as they are analogies -- heaven and shadow represent the popular belief of light vs. darkness, God vs. Satan, etc. Fire and what should more correctly be termed Cold represent the activities of physical energy -- entering and leaving objects, redistributing itself through atomic motion. Those are the basic universal constants -- order, disorder, light, and darkness. But CC does not deal extensively with the metaphysics of the universe at large. Rather, it is more the story of this one planet, and objects interact differently with the nearby influence of an object as large as a planet -- especially if one takes planets, and the life that inhabits them, into account. The green and yellow elements consist of phenomena that only occur on a planet -- lightning, wind, nature-based powers, and the like. They are the colors that make up a planet's individuality, while the other four are universal constants that apply to that planet. Combining all of these ingredients produces the song of a planet -- our fair world has a melody which it sings to the life that can hear it, but it is but one of an uncountable number in the universe, all of which have their own voices to lend, hence the harmony metaphor. As Lavos wishes to consume all of the universe, it seems fitting that the combined power of the universe would stand against him.
So, basically, it boils down to this:
Heat, Cold, Heaven, Shadow -- Universal physical constants, the basis for all of physics and metaphysics in Chrono
(Green, Yellow) -- Strictly planetary phenomena, manifestations of a planet's spirit usable by "lesser" lifeforms
As far as the animals using elements go, these are likely simply denoted as Elements under the CC battle system, when in fact they are, to the animals, natural Tech skills -- the animals, as Leebot has pointed out, must have evolved or been evolved to use these abilities. The most likely explanation for humans using some of the same element abilities is that the Dragonians took the techniques from the animals -- they are extremely connected to the natural world, after all. Seeing as Chronopolis may be at least partly responsible, I move that they would use anything that would increase their power, regardless of its origin. Yes, some of the animals are extremely dangerous, but the Element resources to combat them have been made available to humans one way or another.
Part of this theory is, admittedly, poetic license based on story themes utilized in the series, but in this case, the case of dealing with a fictional universe possessing magic and spirits as one of its variables, that it is at least semi-permissible. Not only that, I do not believe that there is any piece I have left unaccounted for.
On the issue of the Lavos Shell, his facility with magic likely had something to do with the fact that he survived the impact.