Does $\exists x\forall yQ(x,y) \to \forall y\exists xQ(x,y)$? Consider the following examples,

*

*$x$ loves $y$, where $x$ and $y$ are humans

somebody loves everybody $\to$ everybody is loved by somebody


*$x$ can open $y$, where $x$ are keys and $y$ are locks

some keys can open all locks $\to$ all locks can be opened by some keys
But does "$\exists x\forall yQ(x,y) \to \forall y\exists xQ(x,y)$" always hold true?

Yes.
One way to prove this is by using a proof tree. You start with the negation of the formula in question then apply a series of contradiction hunting rules to show that, no matter what, that negation is false, like so:

This tree was generated here.
