Statement 1 is not directly expressible in ZFC, even as a scheme, since you are making explicit reference to the truth of an arbitrary formula $\varphi$. So you need a truth predicate to express the axiom.
But if you have ZFC in the language with a class truth predicate for first-order truth, then I claim statement 1 is provable. That is, it is provable in the theory GBC+$T$ is a first-order truth predicate.
To see this, suppose that $W$ is a proper class of structure, or even any class of size more than continuum. Since there are only continuum many different types in the language of set theory, it follows that there must be at least two distinct elements $X$ and $Y$ in $W$ with the same type. In other words, $\varphi(X)$ if and only if $\varphi(Y)$ for these two objects.
Unless I have misunderstood, statement 2 seems to be provable in ZFC. If $W$ is a proper class structure with signature $\sigma$. Let $Y$ be any uncountable substructure of $W$, and by Löwenheim-Skolem, let $X$ be any countable elementary substructure of $Y$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.