If I so happen to get Kaya the Inexorable emblem, which would be amazing on its own, and I have mutated Lily, the Snapdax, Apex of the Hunt onto a Mindleecher, do I get to cast one from the pile of exiled stuff or is it still in ownership of the Opponent. Even though I know what was exiled because they are facedown and only I can look at them.
The Kaya emblem doesn't allow you to cast a spell from among cards you exiled with Mindleecher, since you don't own those cards. This is so even though you're allowed to look at such a card while it remains exiled. (For ownership, see C.R. 108.3.)
Even though Mindleecher's last ability allows you to "look at and play" cards you exiled with Mindleecher "for as long as they remain exiled", you can play such a card this way only any time you could normally play a card of its type (so not while a spell or ability, such as the emblem's ability, is resolving). This is because the duration "for as long as they remain exiled" indicates a continuous effect (C.R. 611.2a). See also this thread.
With the emblem, you can "cast a legendary spell ... from among [a] car[d] ... in exile" only if you own that card. And if that card is face down, another requirement is that you must be "allowed to look at the face-down card in exile" (e.g., due to Gustha's Scepter) (C.R. 406.3b).
Even though Mindleecher's last ability allows you to "look at and play" cards you exiled with Mindleecher "for as long as they remain exiled", you can play such a card this way only any time you could normally play a card of its type (so not while a spell or ability, such as the emblem's ability, is resolving). This is because the duration "for as long as they remain exiled" indicates a continuous effect (C.R. 611.2a). See also this thread.
With the emblem, you can "cast a legendary spell ... from among [a] car[d] ... in exile" only if you own that card. And if that card is face down, another requirement is that you must be "allowed to look at the face-down card in exile" (e.g., due to Gustha's Scepter) (C.R. 406.3b).