Are all universal formulas true and all existential formulas false in the empty model?

I understand that all formulas with a universal quantifier are true in the empty model and all formulas with an existential quantifier are false in the empty model. However, what about something like, $(\exists y) (\forall x) x=x$ or even $(\exists x)(\forall x)x=x$? I have a hard time believing they are false. And dually, what about $(\forall y)(\exists x)x=x$ or $(\forall x)(\exists x)x=x$? I have a hard time believing they are true. So my question is, is it the case that all formulas that begin with a universal quantifer are true in the empty model, no matter what follows after, and is it the case that all formulas that begin with a existential quantifier are false in the empty model, no matter what follows after?

The mere presence of a universal or existential quantifier is irrelevant. The only relevant thing is whether the statement’s outermost logical operator is a quantifier.

Any sentence of the form $\forall x \phi$ where $FV(\phi) \subseteq \{x\}$ is true in the empty model, and the corresponding sentence $\exists x \phi$ is false in the empty model. This follows from the definition of $\models$.

Note that in general, when we’re deciding the truth of some proposition $\phi$ in a model $M$, we consider the statement $M, u \models \phi$, where $u : D \to M$, $D$ is a finite set of variables, and $FV(\phi) \subseteq D$. In the case of the empty model, the only such $u$ will always be the trivial empty $u$ where $D = \emptyset$.