Once Cosmic Intervention resolves, any “When [this creature] dies” abilities of permanents you control won’t trigger for the rest of the turn because those permanents won’t be put into the graveyard.

If a token permanent would be put into a graveyard, it’s exiled and then ceases to exist. It won’t return to the battlefield.

If you return an Aura this way, you choose what the Aura will enchant just before it enters the battlefield. An Aura returning to the battlefield this way doesn’t target the player or permanent it will enchant, so permanents or players with hexproof may be chosen; however, the chosen recipient must be able to legally be enchanted by the Aura, so a player or permanent with protection from one of the Aura’s qualities can’t be chosen this way. If there’s nothing legal for the Aura to enchant, it stays in exile.

Casting a foretold card from exile follows the timing rules for that card. If you foretell an instant card, you can cast it as soon as the next player’s turn. In most cases, if you foretell a card that isn’t an instant (or doesn’t have flash), you’ll have to wait until your next turn to cast it.

Each permanent that is exiled instead of being put into a graveyard from the battlefield creates its own triggered ability to return itself. Those abilities all trigger at the beginning of the next end step, and you can choose the order in which those abilities will go on the stack. The cards will be returned to the battlefield one at a time as those abilities resolve.

If you’re casting a foretold card from exile for its foretell cost, you can’t choose to cast it for any other alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, those must be paid to cast the spell.

Because exiling a card with foretell from your hand is a special action, you can do so any time you have priority during your turn, including in response to spells and abilities. Once you announce you’re taking the action, no other player can respond by trying to remove the card from your hand.