It goes together with διὰ. διὰ τί = τίνος χάριν = τίνος ἕνεκα etc. Yes, it is an interrogative pronoun.
The implied subject of ἀπέχονται (3rd person singular) is the Λακεδαιμόνιοι of the previous sentence.
Notice also how αὐτοῦ depends on ἀπέχονται in the said sentence. This is true for the sentence you are translating now.
No, it is not.
Check also for different meanings of ἀπέχονται under II. Med.