Probability: Are disjoint events independent?

You are correct, and your reasoning is spot on. Disjoint events aren't independent, unless one event is impossible, which makes the two events trivially independent. Let's take the simplest situation possible as a counterexample. Let $A$ be the event that a fair coin lands heads and let $B$ be the event that the coin lands tails. $A \cap B = \emptyset \implies P(A \cap B) = 0 \ne P(A)P(B) = \frac{1}{2}\cdot \frac{1}{2}.$ The mathematical definition of two events being independent is $P(A \cap B)=P(A)\cdot P(B)$ thus if $A \cap B = \emptyset$ then $P(A \cap B) = 0 \implies P(A) = 0$ or $P(B) = 0$

Two disjoint events can never be independent, except in the case that one of the events is null. Essentially these two concepts belong to two different dimensions and cannot be compared or equaled.
Events are considered disjoint if they never occur at the same time. For example, being a freshman and being a sophomore would be considered disjoint events.
Independent events are unrelated events. For example, being a woman and being born in September are (roughly) independent events.