Would a cell be able to function without a nucleus? why or why not?

YES, however cells with a nucleus cannot survive long-term without the nucleus but can survive for short periods such as during Cloning.

There are 2 types of cellular organisms: Eucaryotes contain a well defined nucleus and Procaryotes do not. So Procaryotes are indeed "cells without a nucleus"

Yes. A cell without a nucleus (that has been made without a nucleus) is called a prokaryotic cell. They lack nuclei or any other organelles that are membrane bound. Most are unicellular and their DNA is held in a nucleoid in the cytoplasm.