Yes, and yes (if you're talking about minors) Otherwise you have to pass the written test before you get a permit - they don't want someone who knows nothing about driving to get behind the wheel.
It's impossible for you to all have seatbelts in a Grand Cherokee in the configuration you have listed - legal seatbelts anyways - unless you have a rear-facing seat in the cargo area. I don't think Grand Cherokees come with those as an option, so to answer your question, yes - it is illegal...
What's the big deal - it fell off - why can't you tell your mom about it without fear of being kicked out??? Maybe your brother didn't install it correctly or the anchor was the wrong type for that application - don't take it personally - it's not that big of a deal.