"They live with their grandfather."
Translation:वे अपने नाना के साथ रहते हैं।
Earlier in this section another sentence (नेहा और पीटर साथ में रहते हैं) uses "के साथ" without the "के". Why is it OK in that case, but not in this one? Can I use "साथ में" in this sentence, like this "वे अपने नाना साथ में रहते हैं"?
You can consider the अपने to be analogous to 'their' in the English translation while के is analogous to 'with'.
But then why is साथ needed. Doesn't साथ mean with? Or in this case do you need के साथ? If that is the case, when is the के needed with साथ and when is it not?
साथ means "together". It also means the noun "side." So you can think of के साथ as "[by] X's side", i.e. "[together] with X". Also, it appears that के is invariable here, i.e. you would never (I think) see का साथ in the sense of "with" (but you could if "side" is meant). This is according to my research; I am not a native or fluent.