Statement 1 is not directly expressible in ZFC or GBC, 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 both expressible and provable in the theory GBC+$T$ is a first-order truth predicate. 

To see this, suppose that $W$ is a proper class or even any class of size more than continuum. Since there are only continuum many different 1-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 1-type, that is, with the same pattern of truths for all the various formulas $\varphi$. 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 substructure of $W$ of size larger than $|\sigma|$, and by Löwenheim-Skolem, let $X$ be any elementary substructure of $Y$ of size $|\sigma|$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.