I like the anime adaptation of "Dr.s Stone". I like the anime adaptation of "Dr. Stone".

I assume you are purely talking about the anime adaptation here, at least in this sentence.

I like the anime of "Dr.stone". I like the anime "Dr.stone".

"the" because we are talking about a specific anime.

I like a Japanese animation of Dr.stone, because this isit because I find the story to be fantastic story. I like it because I find the story to be fantastic.

Rewrote it so it is less repetitious and to emphasize your own personal feelings about the show.

I like athe Japanese animation of Dr.stone, because thisof its fantastic story. I like the Japanese animation Dr.stone, because of its fantastic story.

This story use chemical knowledgChemistry is a key part of the story, which I also like. Chemistry is a key part of the story, which I also like.

Rewrite to make it sound more natural. I also combined the following sentence into this one as I think it flows better this way.

This story uses chemical knowledgestry. This story uses chemistry.

I like that!

See above correction.

Thise anime exist comic versionis an adaptation of a comic book series. The anime is an adaptation of a comic book series.

Combined the following sentence into this one to flow better. I think if you are using the term "anime" you can also consider using the word "manga" intead of "comic" to differentiate it from western comic book series. In that case, the correction becomes: "The anime is an adaptation of a manga series." Note I removed the word "book" as "manga book series" is not natural sounding.

This anime existhas a comic version. This anime has a comic version.

OR There is a comic version

It is original.

Again, see above correction.

This comicIt is one of the most popular in all of jJapanese comic book series. It is one of the most popular Japanese comic book series.

"This comic" is unnecessary as the reader knows what you are talking about due to context. Again, if you are using the term "anime", consider using the loan word "manga" instead of "Japanese comic books". Like Japanese, "manga" in English is both singular and plural, so you do not add a plural s to the end of it. The correction using manga is something like: "It is one of the most popular manga series."

This comic is one of the most popular in all of jJapanese comic books. This comic is one of the most popular Japanese comic books.

Japanese is capitalized. "one of the most popular" ex: My friend is one of the most popular students. This actor is one of the most popular actors.