Well, which word order you would use as a translation is just a matter of emphasis.
More natural or not, it really is quite far from the Polish sentence, too far.
What is the thing with "Swoją"? Why is this applicable to my or his or her? Is it like a magic word that can pinch hit for all except when you need to be polite with a Pan or Pani?
It always refers to the subject of the sentence. If only it is correct in a given sentence, it always is better than the 'normal' possessive.
This course has introduced it too late and therefore there are many sentences which would be a lot better if they used it instead of the 'normal' possessive.
"pan" and "pani" also work perfectly well with "swój".
The together really is not necessary in my opinion, when you work with your aunt, you are working together. It sounds redudant.
Does "razem" have the sense of "at the same time"? I am thinking of "teraz", which looks like "this time" = "now".
Not really, although perhaps in a context specific enough, it could be used for a situation where both you and your aunt work 'separately but at the same time'.
If the context makes it clear that you're talking about your aunt, then "moją" can be omitted here, right?
No it isn't, unfortunately. "Together with" is like a compound preposition, and so goes before the noun to which it applies.