What is the ruling on a person in a state of major impurity (junub) or a menstruating woman (ha'id) reciting the Quran from memory?
It is not permissible for a menstruating woman, a postpartum woman, or a person in a state of major impurity to recite anything from the Quran, whether from memory, from the Quran, from a phone, or a computer. It is also not permissible for them to touch the Quran, based on what was reported from Ali bin Abi Talib that the Prophet (peace be upon him) was not prevented from anything regarding the Quran except major impurity (janabah). (Reported by al-Tirmidhi who said it is a hasan sahih hadith). Menstruation and postpartum bleeding are analogous to major impurity (janabah) as they are all major impurities (hadath akbar).
For those mentioned, it is permissible to mention Allah and supplicate even with verses from the Quran, provided they do not intend them as recitation of the Quran, but intend them as remembrance (dhikr) or supplication (du'a). And Allah the Almighty knows best.
Is it permissible to sleep before the time of the Fajr prayer begins, with the intention of waking up shortly afterward?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible for a Muslim to sleep before the time of the Fajr prayer begins, intending to wake up and perform the prayer, by setting an alarm or asking someone to wake him. What is to be avoided is that a person sleeps while he predominantly assumes that he will not wake up for the prayer.
If he does not wake up until the time of the prayer has ended, he must make it up. It is recommended that he hasten to make it up as soon as he wakes, and likewise recommended that he make up its sunnah along with it.
It is stated in al-Majmūʿ by Shaykh al-Islām Imam al-Nawawī (may Allah have mercy on him): “Whoever was obliged to pray and did not pray until the time passed must make it up, because of the saying of the Prophet (peace and blessings be upon him): ‘Whoever sleeps through a prayer or forgets it, let him pray it when he remembers it.’ It is recommended that he make it up immediately, because of the ḥadīth we have mentioned. If he delays it, that is permissible, because it is reported that the Prophet (peace and blessings be upon him) missed the Fajr prayer and did not pray it until he had left the valley. Had it been required immediately, he would not have delayed it. Abū Is-ḥāq said: If he left it without a valid Sharia excuse, he must make it up immediately, because he was negligent in delaying it.”
It is therefore the duty of every Muslim to keep the prayer in its proper time, seeking help from Allah the Exalted and striving against the self to do so. And Allah Almighty knows best.
Which is better for a traveler: fasting or breaking the fast?
Fasting is better, unless it causes hardship—in that case, breaking the fast is better.
Which is better during the day in Ramadan: reciting the Quran or performing voluntary prayers?
A Muslim should establish a daily schedule for reciting the Quran in Ramadan.
Once they have completed their daily portion (known as a wird or hizb), they can engage in other acts of worship, including voluntary prayers (nawafil). Prayer itself is called "Quran", because a significant part of it involves reciting the Quran.
Allah says: "BE CONSTANT in [thy] prayer from the time when the sun has passed its zenith till the darkness of night, and [be ever mindful of its] recitation at dawn: for, behold, the recitation [of prayer] at dawn is indeed witnessed [by all that is holy]." [Al-Isra`/78]