The cost reduction applies only to generic mana in the costs of spells you cast from anywhere other than your hand. It can't reduce requirements of a specific color of mana.

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.

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.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying, add any cost increases, then apply any cost reductions (such as that of Sage of the Beyond). The mana value of the spell is determined only by its mana cost, no matter what the total cost to cast the spell was.

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.