I'd already watched it before, but I enjoyed it again because the quality of the story and drawing was highvery good. I'd already watched it before, but I enjoyed it again because the quality of the story and drawing was very good.

If something is "high" then it is far from the ground (example: the clouds are high). You can use it as an adverb (highly) to emphasise something (example: I highly recommend that you watch the movie).

I'd already watchedseen it before, but I still enjoyed it again because of the high-quality of the story and drawing was highs. I'd already seen it before, but I still enjoyed it because of the high-quality story and drawings.

In this sentence, "seen" is a little more natural than "watched." If you want to use "quality" and "high," we usually put them together as "high-quality."

Rather, one of the subjects, the destruction of nature, is an urgent issue all over the world, so it seems to me that this movie still inspires people. Rather, one of the subjects, the destruction of nature, is an urgent issue all over the world, so it seems to me that this movie still inspires people.

This sentence is fine as is. But, I think "themes" would be a little better than "subjects," and I might write that "this movie is still relevant" instead of "inspires people" as right now it's unclear why/how they are inspired. Maybe "this movie still inspires people to think more about their environmental impact" would be a little better.

Therefore, the animals in the movie are also supernatural, and another subject is deicide. Therefore, the animals in the movie are also supernatural, and another subject is deicide.

This sentence is okay, but I probably wouldn't use "therefore." "Therefore" is usually used to show cause and effect, but I don't think the meaning of Mononoke is necessarily the cause for the animals being supernatural and one of the themes/subjects being deicide.