Egypt spoke "Egyptian". It's language evolved over time until it was preserved in church tradition in the middle ages as Coptic. The Ancient language was rediscovered and we have a pretty good idea of what it sounded like, but our knowledge isn't perfect.
Sumeria spoke Sumerian, which I believe we've managed to reconstruct. However, it's a language that was preserved mostly in geographic and religious words in Akkadian. Akkadian was the language of the Babylonian and Assyrian Empires. Over time, thanks to the Aramaic invasions and their rise in importance as commercial traders, Aramaic became a common language throughout. Darius I decided it made the most sense to use it as the official language. Old Persian (which isn't technically Avestan, but related) would have been used by the Persians during Cyrus' reign, but Aramaic was official from Darius until the end.
EDIT: I know Montezuma says "Imperio" which is borrowed from Spanish. What others shouldn't he say?