The Thirteenth Doctor's last ability untaps creatures you control with any kind of counter on them, not just creatures you control with +1/+1 counters on them.

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.

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.

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.

Unlike other creature types in Magic that are each only one word, the two words "Time Lord" represent a single creature subtype. Time Lord is the only two-word creature type.

Neither "Time" nor "Lord" are creature types. Some older cards were printed with the subtype "Lord," but all of those cards have updated Oracle card text that removed that type.

If an effect instructs you to choose a creature type, you may choose Time Lord.