How valid is a vow to avoid streets where prostitutes' houses are located, and is it obligatory to adhere to it? What should be done in the event of passing through these streets by car during the day?

Avoiding places of temptation is an act of closeness to Allah and obedience to Him.

Therefore, vowing not to pass through these roads is a vow of obedience and righteousness that must be fulfilled. If the person who made the vow is unable to fulfill it, then expiation for breaking an oath becomes obligatory upon him.

As for passing by car during the day, the breaking of the oath depends on the intention of the one who made the vow: if he intended not to pass at all, then he has broken his oath. But if he intended not to pass during specific times and circumstances, then he does not break his oath by passing at other times. The latter is the more probable interpretation, because the purpose of not passing is to avoid temptation.