Tell your friend if you can. A huge part of what made cutting become a problem for me was the secrecy. No one knew, so it got out of control. Also, I know this isn't what you want to hear, but stopping now will be much easier than stopping weeks, months, or years from now.

As for the issues that led to this, I can relate somewhat. My mom has several health issues as well, and my parents don't exactly get along either.

On how to deal with that, I wish I had some magical perfect advice for you. Honestly I don't because that doesn't really exist. I don't know if you believe this or not, but if you think any of your parents problems are because of you, that is not true. I know parents having issues can be stressful. I don't know how old you are, but there's usually a point where you realize that your parents problems are just that, their problems. You have to live your own life. They both love you, which is what matters most.
Now, your dad's health issues. Do you mind me asking what they are? My mom has dealt with severe arthritis since I was born, and now may have cancer. Each of those is stressful in a different way, so if it's similar to either I may have better advice for you.

As far as your friend, you may be able to settle things between you. I don't know how to help with that (it's amazing I have friends myself lol) other than just tell him how you feel.

Last but not least, what you said about your father and your weight/eating. Don't let it affect how you see yourself. Ask him to stop and tell him why it bothers you. If you want to lose weight, lose weight. If you don't, then don't. It's your life, your body.