Can I name my business Zachary TG designs without registering it?

My name is Zachary T. Gudmundsson, and I live in Ontario, Canada. I was wondering if I can create a business, Zachary TG designs, without registering it. I know I can do this if I call it Zachary T. Gudmundsson designs, but I like the shorter version better. Can I do this without registering it? Also, I am thirteen, so is creating a business legal at this age?