A polyhedron is defined as the intersection of finitely many generalized halfspaces. That is, a polyhedron is any set of the form $ \{x \in R : Ax \leq\ b \} $
I would like to understand this further.
Given that $ Ax \leq\ b, Ax \geq\ b \leftrightarrow\ Ax=b$, a polyhedron can be a hyperplane. Can it be a single point?
Further, in the case that there is no intersection (for example two parallel lines in $R^2$), does the set describe an empty polyhedron, or does it simply fail to define a polyhedron? I.e. can a polyhedron be empty?
It only depends on your convention. A set defined by linear inequalities can certainly be empty. Weather it can be called a polyhedron or not, depends on the convention you choose. It's a matter of terminology, not a matter of `correctness'.