Probably the "question via intonation" was common: "Me lavare potes?"
Maybe orally, but when we write in Latin, no. They didn't have "?" in Latin, so no way to differentiate a question.
Is the -ne or some kind of question word required by Duo for questions?
The -ne is necessary for yes/no questions only.
If you have a "question word" (like ubi , quo , cur , quis , quid , etc.), you don't need -ne. (Note that the -ne is already "attached," in Nonne . Another way of starting a question is Num .)
Me lavare potesne?
I think it's important to signal the question, by putting -ne on the first word in the sentence. "Mene lavare potes?"
Not that's an interested phrase