Does Ashiok, Nightmare Muse's ult allow affected adventure cards to be cast as adventures?
That ability on Ashiok allows you to cast the Adventure part of the card from exile.

When a card that is cast as an Adventure resolves and is exiled, it creates a continuous effect that allows you to cast the creature part, and only the creature part, from exile. This is what it means when it says that "it can't be cast as an Adventure this way". 

Ashiok ability creates a separate effect that allows you to cast that same card. That effect has no restrictions on what types of spells it can cast, so it can cast either the Adventure part or the creature part of that card from exile.