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?
I am guessing possession is typically only specified if "jag har en knapp i din fickan.", et cetera?
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.
Yep, it's the (very roughly) fhw sound. In Swedish there are lots of letter combinations that can make the sound, like sk, skj, ti....
Why would the article be left out? I haven't seen that in any other instances
Why is it wrong to translate it "I eat soup with spoon"? I know it's not the way we say it in English (because we need the "a" before "spoon" but in strict theory the sentence in Swedish doesn't have "en" before "sked"
That's commonly how it works. Compare e.g. "I go to work by car", where you wouldn't say "by a car".
Since there was no article with sked, I didn't think it could be "with a spoon" and tried "by spoon". Why was the article not needed, and how come "by spoon" isn't equivalent?
It's a matter of idiomatic differences only, really - Swedish prefers not to use the article, whereas English does want it. The sentence isn't meant to be translated word-for-word, so we don't allow "by spoon".