Mnemonic Betrayal doesn’t change when you can cast the exiled cards. For example, if you exile a sorcery card, you can cast it only during your main phase when the stack is empty.

You can’t play land cards exiled this way.

Only the cards that remain in exile are returned to their owners’ graveyards. For example, a permanent spell cast this way will remain on the battlefield.

In a multiplayer game, if a player leaves the game, all cards that player owns leave as well. If you leave the game, any spells or permanents you control from Mnemonic Betrayal’s effect are exiled. If you leave the game before Mnemonic Betrayal’s delayed triggered ability resolves, the cards remain exiled.

If you cast a spell this way and that card is exiled, it’s considered a new object. Mnemonic Betrayal no longer allows you to cast it. It remains in exile when Mnemonic Betrayal’s delayed triggered ability resolves.