The Ninth Doctor's ability triggers during the untap step and goes onto the stack at the beginning of the next step (usually the upkeep). The additional upkeep is taken after that step, not after the untap step.

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.