I am trying to find at least one example of rings for every combination of the following properties
Any ideas for the remaining cases?
I see three rows of ???? that you are apparently asking about.
For the first one occurring, you could use the ideal $(x)/(x^3)$ in $F_2[x]/(x^3)$. It is properly inside a local ring, so it doesn't have any idempotents aside from $0$, so no identity is there. It's not a domain since $x^3=0$.
For the second one, the integer quaternions (the subring of the quaternions generated by $i,j,k$ and the integers) (or also the Hurwitz quaternions) will do. They're a domain with identity that isn't commutative.
For the third one, you can take any nontrivial ideal of the integer quaternions.