They are the same question in Danish (as in they use the same words). If you wanted to ask "Is he eating lunch?" then you could add in some sort of time element (e.g. Spiser han frokost lige nu - Is he eating lunch right now?), I think using the continuous constructions here (e.g. Er han ved at spise frokost/Er han i gang med at spiser frokost? - Is he eating lunch?) would be a bit much because to me that would be more long term things you'd use it with, you could also say "Sidder han og spiser frokost?" (Is he sitting and eating lunch?) but whether a Dane would actually say that, I'm not too sure about again (and all of this is covered in a later lesson, I don't think we've added it in this early in the tree just so people don't get confused)

I as a german have a hard time to see any difference between those two questions. I see it now but in german you would need a longer sentence to make clear which of the two you mean.

No. In Danish "morgenmad" means breakfast. "Frokost" used to mean the first meal of the day, but it was typically eaten after a few hours of work, which in modern times would be closer to midday and eventually came to mean lunch.

because the verb here is 'spiser' not 'er' (to be). The verb (spiser/eat) is already conjugated.