What cards can I cast from exile with Dream Pillager?
No, you can't cast any card at all that's in exile that way. You can only cast cards that were exiled by Dream Pillager this turn, and only until the end of the turn.

Dream Pillager's original printed text says exiled this way, referring to the cards exiled by that ability. Its wording was updated with Core 2019 to the following, emphasis added:


  Whenever Dream Pillager deals combat damage to a player, exile that many cards from the top of your library. Until end of turn, you may cast nonland cards from among those exiled cards.


In making this change they were trying to make it extra clear you only get to cast that bunch of cards just exiled, only this turn.