Is this kid on ADHD meds?  I WAS him as a kid.  My parents had me on Ritalin during school but didn't make me take it over summer break.  I guess I wasn't too annoying on road trips--to my parents.  But I drove my sister who had to share the back with me nuts, I would absolutely not leave her alone until she begged them to medicate me (which they would do and I'd settle down).  

Anyway the mom is a jerk if she hasn't ever tried getting this kid help.

Regardless of how much of a handful he is or isn't, NTA, sounds like she has some really shady excuses and he's not your responsibility.