What condition triggers Michael's appearance? He suddenly appeared for me on day 221. I thought it was because I captured the Castle in the west (Lux), but after reloading a save from a week ago, he suddenly appeared the next day on day 146...

In general, he appears on the eighth day after capturing the fort where the souls of angels are imprisoned. But he can also appear on the third day if you conquer the last Angel Castle and destroy all their heroes.

By the way, has anyone received a bonus for defeating him, even if one of his units survives? How did you do it? I bought a Scroll of Earth Elemental Summoning in the artifact shop in the east, defeated all of his helper units except one, petrified it, and then attacked with all my units and defeated Michael. I also considered using the Amulet of Subdual; it also appeared regularly in the shop, as did the Scroll of Earth Elementals. Moreover, I bought both the Amulet and the Scroll, after which no more Scrolls appeared for sale, but a second Amulet did appear, although it's supposed to be a relic artifact.