If you want to prove to your parents that you haven't taken drugs, then there are ways to be tested for them. I'm not sure if it would help in your situation, but I know that my parents would be much angrier if I did drugs than drank alcohol. One thing that you should keep in mind through all of this is that your parents do love you, and what they do is usually for your own good. Maybe they do go over the top every now and again, but you can't blame them for being angry after you had to go to the hospital by your own doing. Yes, they should be supportive and forgiving about these kinds of incidents, but at least they care enough to let you know that they don't want you to do it again.
As for going to live with your friend, I wouldn't advise it. You're only sixteen, and if it's hard for you to handle difficult situations, then staying with another teenager doesn't seem like it would help. Of course, if you think that staying with your friend would relieve some of your stress, then by all means try to what you can, but if you just want to get away from your parents then there could be some negative consequences.
You could tell your parents what's going on in your life, anything you want them to know. It's not always easy to talk to parents face to face about problems, especially when they are the cause of some of them, but if you find a way to communicate what's wrong, then they can try to fix it. If you don't feel like they'll listen to you, then you certainly can write out a letter to them explaining everything, or asking them to do something. Your parents don't want you to suffer, but sometimes they may get caught up in their own lives and overlook the things that you need in yours.
Good luck, and I hope you find a way to have them hear you out.
-Alice