You are 16 and have a job already and do a lot at school. Good for you. Your mom should be happy with you. You shouldn't have to pay rent. As has been mentioned, it is illegal for your mom to charge you rent at 16. As for doing chores...tell your mom that you want to help around the house but have limited time do to work and school. Suggest that she make a set list of chores that you both agree on and that you do them once a week. My mom can be controlling and that is what we do at my house. For my mom it is her mother that is manipulative. (Still is but that's a whole 'nother story.) My brother and I do chores once a week on the weekends since we have time and stick to a set list. (we alternate chores). I am 19 and he is 16 and we have always done chores. However, you should definitely not pay rent. Illegal mean illegal. She is required by law to give you food, shelter, clothes, etc. until you are 18. Try small steps with someone who is controlling. Get her to relax on one thing at a time. My mom is much more relaxed now because of that. (And because I have a somewhat controlling personality too so we butt heads and I wouldn't let her control me so she stopped trying as much). Good luck.
I have a controlling mom too.