Sentence 2 is incorrect and should read 'neither of us were tired' It has the opposite meaning to sentence 1!! Sentence 4 is also incorrect, you can not use either like both here. The only correct way is to use both in that sentence. Either is used to offer choice. You either study Chinese or you study Korean. You can either do as I say or you can go to your room for the rest of the day. Either Paul or Robert can do it today.

"Either" means "one or the other." "Either" (which is always singular, by the way) cannot replace the word "both," which indicates two people or things. "Either of them lives in London" is only possible as part of a larger sentence. Let's see if I can think of something... okay, "You have two sisters, right? If either of them lives in London, ask her to tell the Queen hello for me." Think of "either" and "both" as part of a set of three words: 'both' : one and the other 'either' : one or the other 'neither' : not one and not the other. Unfortunately for you, "either" can be a pronoun, an adjective, a conjunction, or an adverb - look it up in an English dictionary for more examples/explanation.