Bilbo's sharp eyes spot the tiny key. It had evidently fallen out of the pocket of one of the trolls. Otherwise, it would have been turned to stone like their clothes.
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They answer in response to Bilbo's finding of the key by asking.... why on earth didn't you mention it before?!
Durin's Day indicates that the dwarves can open the door with the key on the last moon of Autumn.
Mirth is happiness and joy. Shared laughter and smiles are key elements to mirth, along with song and dancing.
Stand by the grey stone when the thrush knocks, and the setting sun with the last light of Durin's Day will shine upon the key-hole
The door turns visible on a specific full moon every so often. The riddle has something to do with the time the thrush cracks the snail. Bilbo hears the thrush crack the snail, and realizes that today is the day the door can be opened.