Classification, Koszulity and Artin–Schelter regularity of certain graded twisted tensor products

Let $\mathbb{k}$ be an algebraically closed field. We classify all of the quadratic twisted tensor products $A \otimes_{\tau} B$ in cases where $(A, B) = (\mathbb{k}[x], \mathbb{k}[y])$ and (\mathbb{k}[x, y], \mathbb{k}[z])$. determine when a product this form is Koszul, it Artin-Schelter regular.