A triggered ability that triggers when a spell is cast from anywhere other than your hand, such as that of The Thirteenth Doctor, functions only on the battlefield, so it doesn't trigger when you cast that spell from a zone other than your hand.

If a spell or ability allows you to copy a spell on the stack but doesn't specify that the spell is cast, that spell wasn't cast and won't be counted by paradox abilities. However, if a spell or ability allows you to cast a copy of a spell, that spell will be counted for paradox abilities.

If you cast Surge of Brilliance from exile, it counts itself when counting the number of spells cast from anywhere other than 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.

An instant or sorcery that counts how many spells you've cast from anywhere other than your hand, such as Surge of Brilliance, counts itself if it was cast from a zone other than your hand.

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.

Paradox abilities count any spells cast from zones other than your hand. These are usually spells cast from exile, the graveyard, or the command zone. They also count spells cast from outside the game, such as spells cast with Wish or Garth One-Eye's ability.