Does the jurisprudential maxim "Whatever is built upon the unlawful is unlawful" mean that what results from the unlawful becomes unlawful, even if it is legally permissible, such as an unlawful means encouraging a lawful act?

The aforementioned rule does not apply to the examples given, such as listening to music for studying or committing a sin on the road. Rather, it refers to what is built upon something unlawful, like forbidden contracts. Whoever commits something forbidden—such as listening to music for studying—must repent from the forbidden act, but the studying itself does not become unlawful. Similarly, whoever disobeys Allah on their journey should repent from their disobedience, without being obligated to return and cut off the road for a permissible reason.