1st quest: I recruited 180 elves, 16 gremlins, 16 imps, and 541 peasants. I received the key.
2nd quest: I recruited 16 liches, 20 vampires, and 25 zombies. When I approached the quest tent again, my army also included elves, in addition to everything else. I received the key.
3rd quest: This one was very simple - 1 colossus, 2 devils, 2 dragons, 5 angels. As in the previous quest, my army also contained remnants from the previous quests, in addition to the named creatures. And again, I received the key.
At the same time, my father completed the same map in the same way, with the same number of units. He couldn't get the last key. The game says that the quest is completed, but the key isn't given. I think this is some kind of bug.