Once an ability that causes a creature to explore begins to resolve, no player may take any other actions until it's done. Notably, opponents can't try to remove the exploring creature after you reveal a nonland card but before it receives a counter.

If no card is revealed, most likely because that player's library is empty, the exploring creature receives a +1/+1 counter.

If a resolving spell or ability instructs a specific creature to explore but that creature has left the battlefield, the creature still explores. If you reveal a nonland card this way, you won't put a +1/+1 counter on anything, but you may put the revealed card into your graveyard. Effects that trigger "whenever a creature you control explores" trigger if appropriate.

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.