Suppose that a set of sentences of a 1st order language has an infinite model $M$.
Under what conditions is there is a proper classsized elementary extension of $M$?
How does the answer change if we begin with a proper class of sentences?
Suppose that a set of sentences of a 1st order language has an infinite model $M$. Under what conditions is there is a proper classsized elementary extension of $M$? How does the answer change if we begin with a proper class of sentences? 


The answer to your main question is that in ZFC there is always such a properclass elementaryextension. Theorem. In ZFC, every setsized model in a setsized firstorder language has a properclass elementary extension. Proof. This is easiest to see in the case that the global axiom of choice holds, in other words if there is a class wellordering of the universe. So let me first explain that case. Fix the global wellorder and consider any fixed model $M_0$ in a setsized firstorder language. Using the upward LöwenheimSkolem theorem, there is a proper elementary endextension of $M_0$, and we may let $M_1$ be the least such model arising in the wellorder. Continuing transfinitely, picking the least elementaryextension at each stage and unions at limit stages, we may build up an elementary chain $$M_0\prec M_1\prec\cdots\prec M_\alpha\prec\cdots$$ by a definable procedure whose union will be a properclass elementary extension of each of them and in particular of $M_0$, as desired. But my next observation is that in ZFC you don't actually need global choice. If we fix $M_0$, then by the axiom of choice, we may code $M_0$ by a set of ordinals $A$. Consider the inner model $L[A]$, which satisfies ZFC and global choice. Since $A$ codes $M_0$, we may undertake the argument of the previous paragraph inside $L[A]$ to get a properclass elementary extension of $M_0$. In the original universe $V$, then, we get an $A$definable proper class elementaryextension of $M_0$, as desired. QED Your second question, however, can fail in some models. I claim that it is possible to have a definable properclasssized theory $T$ in a model of ZFC, such that every subset of $T$ has a model, and so in particular the theory is consistent, but there is no definable (allowing parameters) model of all of $T$. For example, assume we are working in a model of ZFC in which there is no definable linear order. (I explained how to construct such a model in my answer to Asaf Karagila's question, Does ZFC prove the universe is linearly orderable?.) Let $T$ be the theory of a linear order $<$, with a constant symbol $\hat a$ for every object $a$. Every restriction of $T$ to only setmany constants will have a model, since in ZFC every set is linearly orderable, but in this model there is no definable model of all of $T$, since there is no definable linear ordering of the universe. 

