So am I correct in inferring 'Kinder' turns to 'Kindern' in the dative case? Are there any rules governing which nouns need a declension in the dative?
Yes, there are: http://en.wikipedia.org/wiki/German_declension#Nouns
Why is "Mit welchen Kindern sind Sie?" marked wrong?
why not "Bist du mit welchen Kindern?"
the sentence structure is not correct. But the correct translation sucks as well. I'm German and it should either be: Mit welchen Kindern bist du hier? oder Welches Kind ist deins (which is a free translation and literally means which kid is yours).