I am trying to get an idea about how to classify rings based on the
various properties they can have.
In this regard, I am stumped by the question of whether there exist
rings which are:

Case 1) Non-commutative and have no-zero divisors: I know that Z and
Z_p (p prime) have no zero divisors, but they are commutative. The
only non-commutative rings I can really think of are matrices, but
they can have zero divisors. (In fact, I've tried to either consider
subrings of matrices which might not have zero divisors or construct
various rings based on vectors, like a ring whose elements are vectors
in R^2 or R^3, where + is the typical componentwise addition and * is
either a dot or cross-product. While the cross product, is non-
commutative, it still has zero divisors). My guess is that these types
of structures exist though, but I just don't know enough examples of
rings to identify one that satisfies these conditions.
Case 2) Have unit elements, but are nor groups under *, not
commutative, and have no-zero divisors: No clues as to if this could
even exist. These conditions seem very restrictive and I have to
wonder if this type of structure even exists.

And yes, they do exist.

If you have a ring with no zero divisors in which every nonzero
element is invertible, you have what is called a "skew-field" or a
"division ring". One reason you might be having trouble coming up with
examples is that there are no finite division rings that are not
fields (that is, not commutative); this is called Wedderburn's
Theorem.

A ring with no zero divisors and a unit, but possibly non-commutative,
is sometimes called an "entire ring" (careful, though; some authors,
e.g. Lang, use that term for integral domains).