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. The idea will be 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 unfoldability property.

In fact, it seems to me that your questions are very closely related to how one often thinks about unfoldable cardinals.