I'm sorry for your loss.
The emotional turmoil you're experiencing may, in fact, be due to grief. It's definitely a difficult thing to go through, especially as a teenager. Even a year later, you can be going through the emotional stages. It's a pain that really doesn't truly go away.
Don't classify yourself with a disorder too soon. You could have cyclothymia, bipolar, or borderline, just based on what you've written. However, you may not have anything, and honestly just be going through a really difficult time. Grief and teenage life can do strange things to one's psyche.
Just don't peg yourself with a mental disorder that will stick to you for the rest of your life. It's not pretty, and it just stands to fuel people's ignorance, especially if you're misdiagnosed.
The best suggestion I have for the time being, would be to keep a journal of some sort, but make it honest. Even if it's a sentence or two on how you're feeling through the day. It doesn't have to be a masterpiece. It may seem like an odd suggestion, but the reason I recommend it would be for a couple reasons:
For starters, it'll help you keep track of your emotional state, and what you're thinking during this time. It may help you, and possibly anyone trying to help you, a better understanding of what you're thinking while you're sad/angry/happy/etc. Sometimes it helps to read back and see patterns of what you were thinking during these times, even if it just gives you a better understanding.
Some people also say that a journal is very cathartic and a good release of emotion. Of course, that's personal preference.
If you, at any point, feel calm enough to do so, talk to your family and possibly explain that you're not feeling the best and if you're sorry, say you are. It may not fix how you feel, but it will at least give them a bit of insight and let them feel a bit better.
The only other suggestion I have for the future, is to talk to a counsellor or helpline or some sort. Sometimes confiding in friends is difficult, but confiding in strangers can be easy. If you feel like you're endangering yourself or others with your actions, I would suggest seeking help immediately. The cutting may not be life-threatening, but is definitely an issue that shouldn't be ignored. If it worsens or gets more frequent, seek help.
I wish you good mental health, and hope things work out for you, one way or another.