I don't think so, because "våra" is plural! "Our child's books" would be "Vår barns böcker".
Actually, barn is an ett-word, so it'd be "vårt barns böcker". But your absolutely right in spotting the plural! :D
So the plural for "our"refers to the number of kids rather than the number of books?
The robot pronounces "barns" with a "sh" English sound at the end (not sure what that sound is called in Swedish, "sk?"). Is this the right way to say it?
You're right, it's weird. But r + n create a retroflex consonant to begin with, [ɳ], and after that, the s sound follows suit, so we get [ɳʂ]. So the whole 'rns' cluster becomes retroflex.
There's a little more about these consonant sounds in Swedish wikipedia here https://sv.wikipedia.org/wiki/Retroflex_konsonant
and the English article is very good, with useful sound clips
No, the plural of child is children, and then we add the genitive s after that: children's must be right.
I wrote "books of our children" and it was corrected to "THE books (...)". Shouldn't it be "böckerna" then?