Translation:Their cat has two really cute kittens.
Male cats don't give birth to kittens. Some male cats are surprisingly attentive to their family though. We just had a sentence about a kangaroo with a joey in its pocket and they did not bother to say "un kangourou femelle". So I am wondering if they could have just said "un chat" and we still extrapolate that the cat is female? Here they definitely wanted there to be no mistake about the cat's gender and have "une chatte".
Could the sentence also be "'Leur chatte a deux trés mignons chatons." ?
No, “mignons” must come after the noun as most adjectives do in French and “trés” modifies “mignons” so it must be with it. The number goes before the noun. https://www.thoughtco.com/introduction-to-french-adjectives-1368789 Scroll down at that site, because they do talk about position of adjectives further down.