Is it permissible to have intercourse with one's wife at night, sleep in a state of major ritual impurity (janabah), then perform ablution (wudu) for Fajr prayer in the mosque and recite the Quran, then go back to sleep until 7 AM to perform the ritual bath (ghusl)? And is the prayer required to be repeated despite doing this without an excuse?

What you are doing by praying while in a state of ritual impurity (janābah) is a grave evil and a heinous sin, and you must repent from it. If you were ignorant of the obligation to perform ghusl (ritual bath) for prayer, then this is the most detestable form of ignorance. If you were aware, then the matter is more serious, as some scholars, like the Hanafis, consider one who prays intentionally while in a state of minor ritual impurity (muḥdith) to be an infidel due to their mockery. Shaykh al-Islam Ibn Taymiyyah holds that whoever prays without purification, considering it permissible, is a disbeliever. Prayer is not valid without purification by consensus, based on the Almighty’s saying: {O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles} (Quran 5:6), and the Prophet's (peace be upon him) saying: "No prayer is accepted without purification." What is incumbent upon you is to cease this action, initiate sincere repentance, and make up all the prayers you performed while in a state of janābah. As for reciting the Quran while in a state of janābah, it is prohibited according to the four imams.