Is the following rule correct?
$\dfrac{\Phi, \varphi \Rightarrow \Delta, \psi}{\Phi, \varphi \rightarrow \psi \Rightarrow \Delta}$
I don't think it is. If I pick $\Phi = \{\top\}, \varphi = \bot, \psi = \bot$ and $\Delta = \emptyset$, then the upper sequence is an axiom (since $\varphi = \psi = \bot$) and is therefore valid, but the lower sequence isn't since there isn't anything on the right side to be satisfied ($\top \land \bot \rightarrow \bot \Rightarrow \emptyset$).
What do you think of my answer? Am I allowed to set $\varphi = \psi = \bot$ and call the upper sequence an axiom? Would really appreciate some feedback.
Your argument that the inference rule is invalid is correct, but you can simplify it slightly by setting $\Phi$ to be $\emptyset$ rather than $\{\top\}$.
A sequent can have no premises or no conclusions. Having no premises is equivalent to having $\top$ as a premise and having no conclusions is equivalent to having $\bot$ as a conclusion. This can be justified by the fact that $\bigwedge \emptyset$ is $\top$ and $\bigvee \emptyset$ is $\bot$ and the fact that $\Gamma \vdash \Delta$ is equivalent to $\bigwedge \Gamma \vdash \bigvee \Delta$.
In particular $\vdash$ with no premises and no conclusions is false.
This gives us, using $F$ instead of $\bot$ so it looks a little nicer.
$$ \frac{F \vdash F}{F \to F \vdash} \;\;\text{is equivalent to}\;\; \frac{F \vdash F}{\vdash} \; \text{which is invalid} $$
I think that this inference rule is intentionally similar to the following inference rule, which is valid
$$ \frac{\Phi, \phi \vdash \psi, \Delta}{\Phi \vdash \phi \to \psi, \Delta} \;\;\text{is a valid rule that looks similar to the prompt} $$