We don't accept non-question-form translations of questions (or the other way round). It is quite possible to create questions just by using intonation in Swedish too, so that kind of sentence always has a better translation. You could say Du betalar? – if intonation doesn't seem to be enough, just add Så at the beginning. We use that form slightly less frequently in Swedish than you do in English, but the difference isn't big enough to motivate adding statement forms everywhere.
It would still be betalar, with jag, hon, han, vi or de.
No, Swedish doesn't change the verb like English you pay vs he pays.
Would this sentence formation be the correct way to lightly and informally ask if an invitation to join along is the other party's treat?
(e.g., in English, "You coming?" is met with, "Sure. Are you buying?")
Would "Betalar du?" replace "Are you buying?" in that case, or would one more likely use a different construction?