In some sense $Ord$ is a "proper class" ordinal. Unfortunately the notion of a proper class ordinal is not a straight forward generalization of the notion of "set" ordinals because the proper classes cannot be members of each other. But there are some hopes to define the notion of a "proper class" ordinal using a different view to the usual meaning of a "set" ordinal. For example if we find a property $P$ which is well defined on both sets and proper classes, and then discover a "theorem" like this:
"A set $s$ is an ordinal (in the usual sense) iff $s$ has the property $P$"
Then we can give a "definition" for the notion of a "proper class" ordinal which is compatible with the usual notion of "set" ordinals as follows:
"A class $C$ called a class-ordinal iff $C$ has the property $P$"
As an inexact suggestion in this direction note that the relations $\in$ and $\subsetneq$ are same on set ordinals, i.e.
$\forall \alpha,\beta \in Ord~~~~~\alpha\in \beta \Longleftrightarrow \alpha \subsetneq \beta$
And the relation $\subsetneq$ in well defined between classes. Now the question is:
Question (1): Is there a known generalization of the notion of set ordinals to proper classes?
If the answer of the above question is positive, then:
Question (2): Is there a proper class order type for any proper class well ordering using reasonable assumptions like the Axiom of Global Choice?
If the answer of the above question is positive too, then:
Question (3): What is the order type of $L$ with Godel's well ordering?