As a native speaker, I think this sounds like an acceptable translation. I would report it if it still isn't accepted.

Swedish word order works a lot like English word order. The boy is reading a book next to me would be Pojken läser en bok bredvid mig.
The sentence Över barnen flyger fåglarna is just as poetic in Swedish as its English counterparts Over the children the birds fly/are flying.

The addition of "that is" doesn't really add much besides adding superfluous words. It would also change the sentence in Swedish as well: "Pojken som är bredvid mig läser en bok."

This would not be a direct translation, although it would mean the same thing in context. Also, your translation implies that you already know the boy, whereas the original sentence implies that you only know him as "the boy next to me."

I would translate that as "Pojken är bredvid mig och läser en bok." (Literally: "The boy is next to me and reads a book.") The Swedish gerund, though it exists, isn't really as common as the English and often has to be changed somehow or it'll sound unnatural.

"Reads a book" is a perfectly acceptable translation for "läser en bok". It's possible that you simply made a mistake elsewhere and Duolingo marked "reads a book" as the mistake instead of the real one. The correcting system is a bit wonky at times.