"That is his seventh horse."
Translation:Det er den sjuende hesten hans.
I had the same question. Initially, I had put "Det er hans sjuende hesten."
"Sin" can only refer to an object owned by the subject of the sentence. "That" refers to the horse, not to its owner.
Yes, bot "sju" and "syv" are quite common. You can use either, but be consistent about it.