After doing some research, I can answer your questions.
All of the legendary Pokemon, including Mew, can spawn as a boss on all of the stages on Happenstance Island, (Level 12.)
I looked at a lot of websites and none of them gave the exact chance of a legendary spawning, but it doesn't seem to be too rare.
While it’s not super difficult to catch Legendary Pokemon in Pokemon Quest, you’ll need to wait until you’re on expedition 12-1 or higher. It’s only on mission 12-1 and higher than you’ll see Legendary Pokemon start to appear as bosses. You won’t always see Legendary Pokemon as bosses when you go on expedition, but there’s at least a chance they’ll show up starting at mission 12-1.
When you defeat a Legendary Pokemon boss, there’s a chance they will drop a Mystical Shell item that you can use to make certain food dishes. Because there’s only a chance you’ll see a Legendary Pokemon boss, and only a chance they will drop a Mystical Shell, you may have to run through several expeditions before you get enough Mystical Shells to make the Ambrosia of Legends.
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On these missions, there's a chance a Legendary Pokemoin from the above will appear as the final boss. When this happens, those bosses have a chance to drop a speical item - Mystical Shells.
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On these levels, there’s a chance a legendary Pokemon will appear as the final boss. It’s only ever a chance – so you may need to grind out these missions for that to happen. When you defeat the legendary boss Pokemon, however, there’s a special bonus: you have a chance to get Mystical Shells as a drop.
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It makes sense that there's not a huge amount of information for the chances of these legendaries spawning, as Pokemon Quest isn't a widely popular game. It's probably about 10-25%, because you would have to keep grinding the levels for hours to just get one legendary to spawn, if the percentage was any lower.
