You choose the target of the second ability as it attacks. The number of time counters is checked at the time the ability resolves. If The War Doctor has left the battlefield by that time, use the number of time counters it had on it the last time it was on the battlefield to determine how much damage it deals to that target.
If The War Doctor phases out at the same time as any other permanents, its first ability won't trigger.
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.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Steve Morris
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
If an effect instructs you to choose a creature type, you may choose Time Lord.
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.
