"Are they eating dinner together?"
Translation:Ĉu ili vespermanĝas kune?
Yeah, English is my second language, but I've also learned the breakfast-dinner-supper triad. Depending on your native meal-names Esperanto translations could be a little different, so I hope you marked that as a mistake.
As a native english speaker from the north east of the USA, I'm very farmiliar (and seemingly they are too) with breakfast-lunch-dinner as the triad.
The explanation I've heard of for the breakfast-dinner-supper vs breakfast-lunch-dinner separation has to do with when you eat the biggest meal of the day (dinner). I'm not sure if it's true or just something that I've seen, but it seems like dinner being the middle of the day meal seems to be more common in rural areas where they do more farming.
It should be "vespermanĝon" (accusative)
"vespermanĝi" is a bit more elegant, more esperantesque, so to say than "manĝi vespermanĝon".
No, it's not more elegant and nobody would look down on you for saying manĝi vespermanĝon. These two options are fully equivalent and there's no difference in meaning, nor style; it's up to your preference or native language background which one to use.
It's exactly like with Mia nomo estas and Mi nomiĝas.