Of course, if you meant "at this time, it is what I am doing" (Can't you see, just look at me!). "Don't I swim?" could mean "I do, don't I?" an extra emphasis is placed on the fact that you do swim (regularly, see how well?), rather than that you are swimming now as opposed to doing something else. http://dictionary.reverso.net/dutch-english/zwem http://context.reverso.net/translation/english-dutch/am+swimming
I have seen I am swimming as "Ik aan het zwemmen.", but I am not sure how that form looks negated in a question.
"good" is an adjective. If he wanted her to comment on how good a swimmer he is, then he would ask "Don't I swim well?" Here he is just asking if he does swim, which of course she can see. This is because we are learning more vocabulary with the same verb tense. It will feel more natural when we learn more verb tenses. (Although, yes, some people use "good" as if it were an adverb as well.)
my point exaclty. What you illustrate is bordering on a slang situation. I would expect someone to say that to a woman in 1950 Bond movie, things have moved along, in the context of expression, in the manner that one person might say something to another that they were trying to impress (note the modern lack of gender). So I feel this is not a good answer.
Common isn't a really good measurement because many people say ya and whatcha instead of what are you, but it wouldn't be correct to accept that.
But I don't think aren't is slang/ vernacular or a regional thing. So indeed I think Aren't I swimming should be reported and accepted.
(I'm open for arguments to the contrary)
Ain't would be slang btw and should be accepted in this course.