Battle Mammoth's triggered ability resolves before the spell or ability that caused it to trigger. It will resolve even if that spell or ability is countered.

If a spell or ability an opponent controls targets a single permanent you control more than once, Battle Mammoth's triggered ability will trigger only once. However, if a spell or ability an opponent controls targets multiple permanents you control, Battle Mammoth's triggered ability will trigger once for each of those permanents.

You choose whether to draw a card as Battle Mammoth's triggered ability is resolving.

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.