Is hearing the sound of the Adhan at an unaccustomed time, and perhaps without it actually being called, linked to shortcomings in performing the obligatory duties?

If the questioner imagines hearing the Adhan at an inappropriate time, there is no interpretation for that. However, if he actually hears it and does not attend the prayer, he is falling short in his duty, as congregational prayer is obligatory for an able-bodied man. If he does not pray at all, then his abandonment of prayer is among the major sins. There is a difference of opinion regarding the disbelief of one who abandons it due to laziness, while one who abandons it out of denial is a disbeliever by the consensus of Muslims. A Muslim must fear Allah and perform his obligatory duties such as prayer, fasting, zakat, and Hajj. If he hears the Adhan, he must hasten and prepare for prayer and attend the congregation.