Yes, you can extend the notion of cardinality to classes, and it is consistent that there is only one cardinal for classes (e.g. $V=L$ implies that all classes have the same size), or it is consistent that there are classes which are incomparable (e.g. if you add two Cohen subsets to a proper class of cardinals, the class of the pairs cannot be definably well-ordered and it is incomparable in size with the class of ordinals). If you agree to violate the axiom of choices then there are even more options here (classes incomparable with sets).
But I don't think there is a lot written on this topic, and it's scattered throughout many papers. Some obvious consequences here, some minor mentioning there.
As for being a large cardinal, there are two type of properties to consider here. Small properties, which only affect the sets below the cardinal (e.g. inaccessible cardinals) and large properties, which affect sets which are not smaller than the cardinals (e.g. measurable cardinals). There are also very large properties which affect pretty much everyone in the universe (e.g. supercompact cardinals).
Since $\sf ZFC$ has a very limited access to proper classes, having the class ordinals as a large cardinal of a large property is meaningless. But you can easily have small properties by taking $V_\kappa$ for some $\kappa$ satisfying the wanted property and considering it as a model on its own. If $\kappa$ is a Mahlo cardinal, then in $V_\kappa$ every closed and unbounded set of ordinals includes an inaccessible cardinal.
Note that requiring that every class of ordinals which is closed [and unbounded] has an inaccessible cardinal actually requires less than a Mahlo ordinals and cutting off the universe, because we have less classes to worry about.
Another peculiar example is a Woodin cardinal. Being a Woodin cardinal is having a small property, which itself is a very large property (if $\kappa$ is a Woodin cardinal, it might not even be weakly compact, but $V_\kappa$ is incredibly rich in internally large cardinals). So you can think of the similar case where the ordinals behave like a Woodin cardinal to some extent. Although, I'm not sure why you would do that.