There was another, kind of similar, example in the clothing module - with something like 'I have a button in my pocket'. In Swedish, the 'my' was not said. It was just 'in the pocket' and you assumed that it was yours. Is this spoon example the same? Could it also mean 'with my spoon'? Or is that only correct if it's 'the spoon' in Swedish?
Yes, and in this case specifically it's more like Swedish has a set phrasing which English doesn't. If English had used the word "by", it'd have done the same: "I eat soup by spoon" without an article and without a possessive. Swedish does the same thing with med here.
No, there's a typo acceptance system, but I'm afraid that's a problem with virtually any text-based language learning course. It's a shame, but it's a difficult problem to solve - and it needs to be solved in a systematic basis, not by contributors in a specific course.