What should we call the 'sets' which don't exist under certain set theory axioms? For example we know that the set of all ordinals does not exist in ZFC, so what should we call it? Set? Collection?

The usual term is proper class.
$\color{red}{\textbf{Warning:}}$
Beware of thinking about proper classes as objects. No model of $\mathsf{ZF(C)}$ has a proper class as an object, and so any arguments which treat them as sets are informal. 
If $C = \{ x : \phi(x) \}$ is a class, it's common to write $y \in C$ when what we really mean is $\phi(y)$. So when we write $\alpha \in \mathbf{On}$ (or similar) to denote the fact that $\alpha$ is an ordinal, what we mean is that $\alpha$ satisfies a particular formula that defines all ordinals (and only ordinals).
There are set theories that have proper classes as objects, e.g. $\mathsf{NBG}$, in which case you can treat them as objects which have members; but in $\mathsf{ZF(C)}$ you can't do this.
Note in particular that, in $\mathsf{ZF(C)}$
$$\{ x \, :\, x \not \in x \} = V$$
where $V$ is the 'universe', the proper class of all sets. But we don't get the contradiction of Russell's paradox: the question of $V \in V$ or $V \not \in V$ does not arise because $V$ is not an object in the model.
