A very nice collection of questions.
Here are a few things one can say to get started.
If $\kappa$ is a measurable cardinal and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, suppose that $M$ is a model of $T$ in which $\leq$ is a well-order and $M$ has size at least $\kappa$. Since $\kappa$ is measurable, we may by iterating the ultrapower maps find an elementary embedding $j:V\to N$ into a transitive class $N$ with critical point $\kappa$ and $j(\kappa)$ as large as desired. So $j(M)$ is a model of $T$, with the order $j(\leq)$ of size at least $j(\kappa)$ and a well-order in $N$ and hence also still a well-order in $V$.
One can do it with less than a measurable. Specifically, if $\kappa$ is merely an unfoldable cardinal (this is consistent with $V=L$, so much smaller than measurable, but above indescribable), and $T$ has a well-ordered model of size at least $\kappa$, then it has arbitrarily large well-ordered models. To see this, note first that a downward Löwenheim-Skolem argument shows that $M$ has well-ordered models of size exactly $\kappa$. Now, we can place this model into a transitive model $N$ of size $\kappa$, and then apply unfoldability to get elementary embeddings $j:N\to \bar N$ with $j(\kappa)$ as large as desired. The same reasoning as before shows that $j(M)$ is a well-ordered model of large size.
You can get a kind of converse from the same idea, using the extension property characterization of unfoldability, if one allows the language to become larger.
Theorem. A cardinal $\kappa$ is unfoldable if and only if every theory $T$ of size at most $\kappa$ with a model of size at least $\kappa$ in which $\leq$ is a well-order, has well-ordered models of arbitrarily large size.
The idea is to write down the diagram of $\langle V_\kappa,\in,A,\leq\rangle$, where $\leq$ is a well-order of $V_\kappa$ and $A\subset V_\kappa$, and then get arbitrarily large extensions $\langle N,\in,A^*,\leq^*\rangle$, which will give you the extension version of the unfoldability property.
It seems to me that your questions are very closely related to how one often thinks about unfoldable cardinals.