Is it permissible to use an elevator that plays music upon reaching a floor, with no intention of listening and with inward disapproval, and knowing that the sound of the music might mix with the Qur’an or the Adhan? And is listening to the Qur’an in the apartment problematic due to the possibility of it mixing with music sounds from the elevator or from neighbors?

The presence of music outside the home does not prevent playing the Quran inside the home. Deliberate listening to music is what is forbidden, but unintentional hearing incurs no sin. This is supported by the hadith (definition) of Ibn 'Umar (may Allah be pleased with both of them) regarding his hearing the sound of a shepherd's flute and placing his fingers in his ears. It is also supported by Ibn Rajab's statement that the act of someone who merely hears, as opposed to actively listens, is not described as forbidden, and that commands and prohibitions relate to active listening, not mere hearing. The best course of action is to plug one's ears when hearing music. If that is not possible, then one must mentally turn away from it with the heart. There is no harm in using an elevator if avoiding it causes hardship, provided one mentally turns away from the music with the heart.