Why does this sentence use है and not हैं? Isn't बच्चे plural?
You're right. It should use Hain. Report it.
Why do we need oblique case here? Could we use उसके instead of उनके?
Yes you can use उसके instead of उनके. That would be similar to using तुम instead of आप. उसके is informal and used for people younger than you. For your elders, even if you're very close to them, you should still use आप and उनके.