How do you know it's "our" Aunt. There is no word for this in the sentence. I'm guessing it's just context related. If it was someone else's aunt you would say "ich ciocia" or something like that.
Yes, you are right. We often assume (and it's most common with family members), that the object of the sentence 'belongs' to the subject. Here it's a bit the other way round, but if some aunt brings 'us' breakfast, it's probably 'our' aunt. What is also likely is that it is 'my' aunt because perhaps the other person of 'we' is my friend, or my girlfriend, or my husband - but still, from my perspective, it's "ciocia" and in a context that will be obvious.
Are sentences with an invisible, implied possessive pronoun - e.g. "(My/Our) aunt is bringing us breakfast" - common in Polish? I experimentally introduced one in another translation exercise, and it was indeed accepted as correct.
Yes. In fact some of them will work even in English, right? "Mom brings us breakfast". Maybe even "Auntie brings us breakfast"? Not sure here, but it seems more likely than with "Aunt".
If in the context it is totally clear who this "Aunt" is and whose aunt is she, that would be totally redundant to mention it.
I think that the word drop-down hints are to some degree independent of surrounding words.