Is it permissible to listen to recorded Quran while doing household chores, even if one is sometimes distracted by thoughts or conversation?

There is no harm in listening to the Quran from a recording device. It is sunnah (definition) to listen attentively and ponder its meanings. If one becomes preoccupied due to a valid excuse, there is no objection. If there is a need to speak during the recitation, the recorder should be turned off, or one should speak in a low voice, based on Allah's saying: "O you who have believed, do not raise your voices above the voice of the Prophet or be loud to him in speech like the loudness of some of you to others, lest your deeds become worthless while you perceive not" [Al-Hujurat: 2], and His saying: "So when the Quran is recited, then listen to it and pay attention that you may receive mercy" [Al-A'raf: 204].