Someone wiped over his shoes after wearing them in a state of purity, then took them off and prayed without them?
Ablution is not invalidated by taking off leather socks or shoes after wiping over them. However, whoever takes them off after wiping must wash his feet only. If he prayed without washing his feet, he must wash his feet and repeat the prayer. And Allah the Almighty knows best.
Does burping (belching) break the fast?
Burping does not break the fast. However, if anything comes up from the stomach, the fasting person must spit it out and rinse their mouth to cleanse it.
If they swallow it intentionally while being able to expel it, or if they swallow their saliva before rinsing their mouth, their fast is invalidated.
Is it permissible to sleep after the time of the Fajr prayer has begun, 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.
If a person sleeps after the time of Fajr has begun, with the second adhān, then sleeping is forbidden (ḥarām) at that point, unless he is certain that he will wake up and perform the prayer within its time. And Allah Almighty knows best.
The Prophet ﷺ said: "There will emerge from my nation people who will drink the Qurʾān as they drink milk" — is this ḥadīth authentic, and what does it mean?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
ʿUqbah ibn ʿĀmir, may Allah be pleased with him, narrated that the Messenger of Allah ﷺ said: "There will emerge from my nation people who will drink the Qurʾān as they drink milk." (Reported by al-Ṭabarānī in al-Muʿjam al-Kabīr, and cited by al-Haythamī in Majmaʿ al-Zawāʾid, who stated: "Its narrators are trustworthy.")
The meaning of this ḥadīth is that there will emerge from this blessed nation people who recite the Holy Qurʾān without reflecting upon its meanings or pondering its rulings — the noble verses passing over their tongues just as milk passes over them when it is drunk, leaving no trace of thought, understanding, or contemplation behind. [See: al-Taysīr bi-Sharḥ al-Jāmiʿ al-Ṣaghīr,Vol.2/P.63] And Allah the Almighty knows best.