AI cheating on single-player maps.

I had an interesting moment when, on a "Heroes of Might and Magic, but not Magic" map with a complete ban on magic, the enemy AI somehow used a magic arrow in one battle.

As far as I remember, magic isn't completely banned on that map. Specifically, the author of this map forgot to remove some heroes' "native" spells from their spellbooks, so when hiring certain heroes, they already know one of their native spells.

Yeah, sometimes there are absurd bugs like AI flying or walking on water.

I've noticed that too, but it seems like we'll just have to accept this fact. :)

Regarding the Grail, the computer doesn't know where the Grail is hidden. It searches for it based on fragments of the map, as evidenced by the many holes where the computer digs for the Grail.