Can we treat proper classes in a well-defined way that has them behave as ordinal numbers?

(1) Can we treat proper classes in a well-defined way that has them behave as ordinal numbers, where we can have arithmetic and some new notion of cardinality unique to proper classes, going higher and higher? (2) Can we do such within surreal numbers as a proper class? (3) If so, can we, like we do with ordinals, imagine infinitely more proper classes, that might aid in realizing the scope of how far we can make extendable the surreals, and proper classes themselves for that matter, indefinitely, beyond a notion of proper class?

To complete Mark S.' answer, there are (say in NBG set theory) proper classes equipped with a well-ordering, which contain a proper initial segment isomorphic to the class $\mathbf{On}$ of ordinals.

For instance take the class $\mathbf{On}\times \mathbf{On}$ of ordered pairs of ordinals, with the lexicographic order. So $\mathbf{On} \times \{0\}$ is an initial segment of $\mathbf{On}\times \mathbf{On}$.

But since $\mathbf{On}$ cannot lie in a class (because classes only contain sets), there is no "generalized" ordinal which is isomorphic the class $\mathbf{On}\times \mathbf{On}$.

I don't know if there are ways to generalize the notion of ordinals so that for instance there exist a unique ordinal isomorphic to $\mathbf{On} \times \mathbf{On}$ as previously defined.

A set theorist may have more to say, but here's my understanding:

(1) The only proper class that's like an ordinal would be the class of all ordinals. And then you can't take the successor. If you wanted to define cardinality as "the least ordinal such that..." then you only get one more "cardinality" for proper classes from this.

If you have something like the axiom of limitation of size then that one new cardinality is enough for all the proper classes. But if you don't have such an axiom, then you could end up with proper classes that the class of all ordinals doesn't "count", even if you had the axiom of choice at the level of sets.

(2) I'm not sure what you mean, but you could put the ordinals (as surreals) in the left position to form a "gap" greater than all surreals.

(3) As in the answer to (1), it doesn't really make sense to go any farther.