2 greek-related questions

[Tellos Athenaios]

Now that I think of it again, probably the Athenians themselves would call it Αθηναίων Ηγεμονία (Athenaion Hegemonia - Athenians' hegemony). IIRC, this is what they actually called the fifth-century Delian League.
Democratic Athenian citizens would hate words like empire, kingdom, etc.

Well there's to say that Arche is first and foremost 'rule'; it means Empire in a 'manner of speaking' sense. But yes, the Athenaioi would likely refer to their role of first violin in the Greek (when they have it again) world as Hegemonia (leadership).