Inspired by this question, I've been trying to figure out for myself the basic properties of definable class well-orderings in transitive models $M$ of ZFC: What is $\omega_1^{CK}(\mathsf{Ord})$?

There were a couple things I couldn't figure out.

- Is it necessarily the case that $M$ is correct about well-orderings? I.e., if $M \models ``\varphi(\cdot, \cdot)$ is a total ordering of $Ord" \wedge \forall x(x=\emptyset \vee \exists y \in x(\forall z \in x(\varphi(y, z)))),$ then $\varphi$ truly (in $V$) defines a well-ordering of $Ord^M?$
- If $\varphi_1$ and $\varphi_2$ define in $M$ isomorphic well-orderings of $Ord^M,$ is there necessarily a definable isomorphism $\psi(\cdot, \cdot)$ between them? E.g., if $\alpha$ is the first ordinal to exceed a proper class of ordinals under $\varphi_1$ and $\beta$ is the first such ordinal under $\varphi_2,$ then we would have $\psi(\alpha, \beta).$

(1) is easy to prove in the case $Ord^M$ has uncountable cofinality in $V,$ but it isn't obvious to me whether it can fail in models of countable cofinality height. (2) seems unlikely, since I don't really see how to extend transfinite recursion past the ordinals (especially in light of potential failures of (1)), but I have trouble imagining what a counterexample would like.