Am I being exploited by my supervisor?
Yes you are.

Point 4 is downright abuse of power. You are not in any way obliged to provide free tuition to your advisor’s child! If your institution has reasonable management, you reporting this would result in severe disciplinary action (I think I would have my tenure clock pushed and have something on my record for something like this). Even if you were paid to do it I’d feel weird about having my students tutoring my kids. Depending on where you are on the planet, not getting paid for tutoring is illegal. Your advisor would be in serious legal trouble if you are in, say, the US or Europe.

Drafting emails seems like an odd practice. It’s abusing power to some extent but if the emails are relevant to your work it might be construed as acceptable by some department heads. That said, you’re a student, not a personal assistant, seems like a waste of your time. 

You should not be reading and correcting others' work with no credit, this is the advisor’s job. I would be upset if I were the one of the students who didn’t get their advisor’s feedback! 

Being an ass about authorship is another bad sign, though in itself may not be too bad. Disagreements about authorship do occur, though the response you describe seems wildly disproportionate. Getting students to the point where they have to 'beg' for your forgiveness is never okay. His job as an advisor is to educate, teach and advise - you are not his pawn or slave. 

Taken together, what you’re describing is a toxic and abusive relationship. 

What you can do depends on where you are in the world and your department. Most reasonable universities have ombudspersons or student support centers whose job is to help people in your situation. If no such thing exists you can try to approach your department head or dean. 

You need to have clear-cut evidence. Emails, records of your visits to tutor the advisor’s kids, written proof that you corrected the thesis of others (say, an email exchange, files sent to you with request for correction, or any other acknowledgement) etc.

I’m sorry you’re in this situation, hope you get things settled. The tone of your post does not make it sound like you are in severe emotional distress, but of course it's impossible to tell. Make sure you get emotional support through what is undoubtedly a rough experience. You do not deserve this, and you should not feel like this is normal.