Since the intersection of any set with $\emptyset$ is $\emptyset$, it does not seem like $\phi$ has any limit point. Is my reasoning correct?
Empty set has no limit points since no set can have non-empty intersection with $\varnothing$.
Suppose $\emptyset$ has at least one limit point, say $x$. Then there exist a sequence $(x_n)$ in $\emptyset$ which converge to $x.$ That is each $x_n\in\emptyset.$ This gives if empty set has a limit point it is not empty which contradicts to its definition. Hence??
Yes you would appear to be correct. And since it contains all its limit points (vacuously), it's closed. Additionally, this appears to be true in any topology.