There are a few cards that allow you to exile cards and then cast them later (Narset, Enlightened Master for example). After a spell cast this way resolves, does it go back into exile, or does it resolve normally including being placed in its owner's graveyard? (assuming nothing else effects where it would go after being cast)
What happens to a spell after it's been cast is not affected by where it was cast from. It's the usual rule about cards being by default "reset" whenever they move from a zone to another.
Unless the spell itself (e.g. Pull from the Deep, White Sun's Zenith...) or another card (such as Rest in Peace) says it doesn't go to the graveyard, that is where it will end up.
If a card is exiled and played, then when it dies it will go back to exile,this is because when its exiled it is considered to be "out of the game" but it is still usable if you cast it, this is just the same as Spawnsire of Ulamog's ability or the affect from Coax from the Blind Eternities.
2This is incorrect. A card doesn't care that it came from graveyard, or hand or exile unless it's ability says it does. When a spell is cast from exile, that spell goes to graveyard, unless it otherwise says to exile it. If you cast say Skittering Invasion from your sideboard for Spawnsire, it goes to graveyard. Creatures go to graveyard when they die unless they are told to exile, like with unearth. Jan 13, 2018 at 20:38
Praetors grasp exiles a card facedown and you may look at and play that card as long as it is exiled what would be the point of this card if after playing it it is just put in the grave. Then it would say play that card then return it to its owners graveyard Jan 13, 2018 at 20:54
3No. once you cast it it leaves exile, going on the stack, the fact that it goes back to exile after you cast it does not change the fact that it has left exile. Jan 13, 2018 at 21:20
1Also exile is not considered to be outside the game, this is why Coax refers to both outside the game and in exile separately. The exile zone is a zone of the game, what is considered by the game to be "outside the game" is almost always the sideboard, in casual outside the game can be anything the players agree on.[IE everything you own] Jan 13, 2018 at 23:16
2@Andrew I think it is the confusion from old cards mentioning "removing from the game" = Exile and the concept of "outside of the game", meaning sideboard or in the case of spawn of Ulamog, your trade binder if you wanted to (in casual play) Mar 13, 2019 at 9:36