If ZFC has a model, must it be at least a countable model? (1) must ZFC have an infinite model? 
(2) if so, why? 
(3) is it because of the replacement schema? 
(4) if so, is it because we have a finite language and so we can only satisfy or describe countably many instances of replacement? 
(5) assuming "yes" to question (1), am I right to say that by Skolem's Theorem, ZFC must have at least one countable model?
 A: The result (existence of a countably infinite model) has absolutely nothing to do with the fact that ZFC is not finitely axiomatizable. Precisely the same result holds for any theory (over an at most countable language) that has an infinite model.  In particular, precisely the same result holds for NBG.   
There is indeed a countably infinite number of instances of the axiom scheme of replacement. That has no direct connection with the existence of a countably infinite model. 
And ZFC can only have infinite models. One needs very little of ZFC for this, not the Axiom of Infinity, not even Powerset.    
A: If ZFC has a model it would have to be infinite. This can follow, as said from power set or the infinity axiom. Furthermore the language of set theory has only one binary relation $\in$, so any theory would be countable and therefore if there is an infinite model there would have to be a countably infinite model.
All this was said before, but I would like to add on an important point:
Even if $\frak M$ is a countable model of ZFC, internally it is a proper class. That is to say, there is no $f\in\frak M$ such that $f$ is a bijection between $\omega$ and $\frak M$.
This model, along with a function witnessing its countability live in a larger model of some strong-enough-theory (this larger model may be a class model).
Note that this has nothing to do with countability. Every set-model of ZFC would think of itself as a proper class, but we "know" (externally) that it is only a set, and if this set happens to live in a universe of ZFC then there is some function from an ordinal (which may be an element of this set-model) onto that model. This should be a hint of how complicated and convoluted infinite objects can get.
