I've entered my proof on this website and I don't understand why following proof isn't okay.
As you can see, the proof checker tells me that I've used the rule for $\exists$ in a wrong way. Is this just a made a small spelling/definition mistake or am I using the rule entirely wrong?
As far as I'm aware I can conclude $\exists x (P(x) \rightarrow Q(s)$ from the subproof via Exist-Elimination if i've written the same thing inside of the subproof.
The problem with your existential elimination attempt is that the witness variable, $s$, occurs free in the conclusion line 5: $\exists x~(Px\to Qs)$.
The argument as a whole is invalid.
$\forall x~\exists y~(Px\to Qy)$ and $\forall x~Px$ are satisfiable exactly when everything satisfies $P$ and something satisfies $Q$.
However, given that, the conclusion $\forall y~\exists x~(Px\to Qy)$ may be unsatisfied when something else does not satisfy $Q$.
Therefore the premises do not logically entail the conclusion you wish.