My sister moved in with us when she was 12, and not because she was having a hard time (my mum was a single parent and died) so it was a little easier for us. But I still struggled. It's hard raising a teen that hasn't been moulded into your family habits from birth. And it's hard putting the mum hat on when dealing with your sister.
As for a chore chart I sat us all down and said "I'm not coping. I feel like we're not all working together and the house is becoming a mess". We made a 4 week chore roster that rotated some jobs and it was for ALL of us. We were all equally involved in discussing who did what jobs, what amount of work was fair etc. I was mindful to make her busy teenage life sound like hard work and say "do you think you could fit in xyz on the weekend?"and things like that. I empathised that she still needed homework and leisure time.
As for family time, my sister and I started walking the dog together when I got home from work. We both really enjoyed and benefited from the time together. I also made Friday take-away night and we would sit at the table, chat and play the "week" game. Warning, I made it up and it's uber daggy LOL we would say "Saturday" and go around the table with each person having to say one thing they did on that day. As we worked through the days of the week it helped us talk and reflect on what had happened and touch base on things we might have beentoo busy to do at the time.
Goodluck. You're doing an amazing job :)