assalamu alaimkum, i have a question that has been on my mind for a while and i think that i've been getting mixed answers. my question is, is it ok to listed to the Qur’an while being distracted, i.e while falling asleep, cleaning ur house, or doing homework. i have been told that it's not ok to do that so i've replced that need with listening to music, but i would prefer to listed to Qur’an, it helps me memorize and helps me sleep better at night. please let me know what you think

All praise be to Allah, and may His peace and blessings be on the greatest messenger, Muhammad.
You must be attentive to the Qur’anic recitation as much as you can. Allah says,
“وَإِذَا قُرِىءَ الْقُرْآنُ فَاسْتَمِعُواْ لَهُ وَأَنصِتُواْ لَعَلَّكُمْ تُرْحَمُونَ”
“So, when the Qur’an is recited, listen to it, and be silent that you may receive mercy.” (7:204)
Many scholars interpreted this to mean listening to the Qur’an during the prayer for those praying as ma’moom. Some say it is in reference to the Qur’an recited during the Jumuah (Friday sermon), while others maintain that it applies to the Qur’anic recital in general.
In tafseer al-Jalalayn, it says,
“This was revealed regarding refraining from speech during the sermon, which here has been expressed by [the recital of] 'the Qur’an', because it comprises it; others say that it was revealed regarding the recitation of the Qur’an in general.” And in Asbaab an-Nuzool (Reasons of revelation) by al-Wahidi, it says,
“… Abu Hurayrah said regarding this verse: “This was revealed about raising voices while in prayer behind the Messenger of Allah, Allah bless him and give him peace”. Said Qatadah: “When the prayer was first obligated, people used to speak during their prayers. A man would come and shout to his friend: ‘How many units of prayers have you performed?’ and his friend would respond: ‘such-and-such’. And so this verse was revealed”. … Sa‘id ibn Jubayr, Mujahid, ‘Ata’, ‘Amr ibn Dinar and other commentators of the Qur’an are of the opinion that this verse was revealed about listening to the prayer leader upon giving the sermon of the Friday prayer.”
From the above, it seems that there are three opinions regarding what is meant by the verse. Yet, it is fair to say that what matters is the immortal statement of the text and, thus, we must be as attentive as possible when the Qur’an is recited in general.
Being busy working or falling asleep while partially listening and partially benefiting is not an insult to the Qur’an, for you are still benefiting from it as much as you can. Such was the fatwa of Sh. Salih al-Fawzaan regarding a woman who listens to the Quran while in the Kitchen cooking. [al-Muntaqa min fatawa al-Fawzaan, Question # 437].
It would be an insult if people are laughing and joking around, and having the Qur’an play in the background while completely distracted from it.
Also if some had the Quran playing in the market or so, you are not obliged to stand there and listen.
If having Qur’anic recitation in the background will help you with all of what you mentioned, then I would certainly advise you to do so. I would also advise that, if your main language is not Arabic, to get the audio recordings of the translated Qur’an to listen to the Arabic text and the translation, which will help you learn what Allah is addressing you with.
This is even more warranted if it will spare you listening to music, which is forbidden in Islam (according to the vast majority) for various proofs, including the following hadeeth,
“لَيَكُونَنَّ من أُمَّتِي أَقْوَامٌ يَسْتَحِلُّونَ الْحِرَ وَالْحَرِيرَ وَالْخَمْرَ وَالْمَعَازِفَ…” “There will come people of my ummah who will make permissible fornication, silk, wine and musical instruments…” (al-Bukhari) I ask Allah to make the Qur’an the life of your heart, the light of your breast, a departure for your sorrow and a release for your anxiety.