I said "De kindered zien hun", which it interpreted as a typo of "hen". Why do I use "hen" instead of "hun" here?
Hen is used as a direct object and after a preposition, hun is used in all other cases. "They" are being seen by the children, making hen a direct object.
Yes, you could say that. It's a tad awkward, though.
Do note that hen is not very common in (colloquial) spoken Dutch. Most people substitute it for ze or (against official rules) for hun:
Ik zie ze.
The above sentence is much more common than the one in the exercise in spoken form.
Does hen here mean the plural they or the singular they (like The Swedish Hen)?