So, could this mean "have you been paying for the food?", or that would be a different verb form?
That's correct. Just like we don't make a difference between "pay" and "is paying", the same reasoning goes for any other applicable verb form as well.
Is mat/maten not translatable as (the) meal at all?
No, that's måltid.
I didn't try it, but would "have you paid the food" also be an ok translation? Or would the "för" be left out in Swedish for that meaning?
Is that really natural English?