Does intercourse lead to not waking up for Fajr prayer, and is that considered a sign of God's wrath?

There is no sin upon you for having intercourse with your wife whenever you wish. If you desire to sleep before Fajr, you should take the necessary measures to help you wake up, such as setting an alarm or appointing someone to wake you. If you do so, but sleep overwhelms you, and you wake up after the time for prayer has passed, then pray upon waking, and there is no sin upon you. Your sleeping through the prayer at that time is not a sign of Allah's wrath upon you, for the Prophet, peace and blessings be upon him, said: "There is no negligence in sleep." If you are keen to have intercourse a sufficient time before Fajr to allow for sleep, you will wake up for prayer, Allah willing.