Yes. "obok" and "koło" are already quite close, but "przy" is, like, literally next to. And with the table, it should mean they are politely sitting on the chairs at the table.
So in the context of a table, eating merely near it is enough to qualify as English "at."
but this could also mean that they eat their meal next to the table? i.e. on the ground
Technically, I guess so. Although I'd use 'koło' or 'obok' in order not to be ambiguous.
Since when is lunch dinner? Or vice versa? The preposition isn't even the issue.
Depends on the dialect. Some natives consider the three main meals to be: breakfast/lunch/dinner, and some use breakfast/dinner/supper.
And in Poland we usually learn the second option. But the first one is in the main answers.
Old European dinner is eaten at noon, supper is eaten in the evening.