I believe the answer to your question is no, you don't have to believe in God if you believe in demons. I was in the same situation when I first realized that I was an atheist because of numerous "demon-like" experiences in my family. Then I began to learn about the more scientific explanations for these things, of course, which satisfied me that there was reason to doubt my belief in the supernatural. But note that I was an atheist first, then skeptical about demons second, because of family related experiences such as what you may have had.
The reality is that you don't know what is causing the phenomenon you believe to be demons. It could be space aliens in invisible suits. It could be time traveling teenagers from the future who snuck back into the past to mess with us who also have some advanced technology. Or maybe they're pan dimensional super beings that exist in numerous realms all at once so they can touch us but we can't get at them. All of these things could possibly do the sort of stuff a person might see as demonic. You would never know the difference. It is, at best, something you can't truly explain. It says nothing about God's existence one way or another.
Also, note that some religions are atheistic. That is, they don't believe in a supreme being like God, but they still believe in the supernatural. This includes some forms of Bhuddism and Satanism. Atheism is nothing more than the lack of belief in God. Most atheists are actually agnostics by the usual way these words are used--atheists who've read about atheism usually disagree with these uses, but that's another story. Technically, when people in the U.S. and other Western countries refer to atheists, they are usually referring to skeptical-secular-atheist types like myself. But these 3 things are certainly consistent with one another, they don't automatically go together. So it sounds like you're an atheist with a belief in the supernatural. There's nothing bizarre about that. Don't let the usual meaning behind these words in normal conversation confuse you.
IsaacJ