Because the card in exile becomes foretold, you can look at it in exile. (Most effects that have you exile a card face down don’t allow you to look at the card in exile.)

The reduction of the foretell cost applies only to generic mana in the foretell cost. It can’t reduce requirements of specific colors of mana. For example, if you foretell a card in your hand with mana cost {1}{G}{G}, its foretell cost will be {G}{G}.

If the card you exile has foretell, it will have two foretell costs, the one printed on it and the one given to it by Ethereal Valkyrie. You may cast that card for either of its foretell costs.

If you foretell a modal double-faced card, the foretell cost will be based on the mana cost of the face you cast from exile. For example, if you foretell Kolvori, God of Kinship/The Ringhart Crest, you could cast Kolvori by paying {G}{G} or The Ringhart Crest by paying {G} on a future turn. If you foretell a modal double-faced card whose front face is a nonland face but whose back face is a land face, you can’t play that card as a land.

Although the card you exile becomes foretold, you did not foretell it. An ability that triggers whenever you foretell a card will not trigger.

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.

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.

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.