Do I need to do the Honours Research Paper or Honours Thesis if I want to apply to graduate studies?

Not necessarily. What you need is a writing sample that you can send together with your application. Different programs expect different kinds of samples, but in most cases, you need to show that you can conduct research independently and that you write well. An Honours Research Paper or an Honours Thesis is a logical way to produce such a writing sample, and it might otherwise strengthen your application and give you useful research experience, but it is not strictly necessary. Many students have been successful without writing one.