Translation:The Swedish chef treated us to lunch.
In the very specific case of meals and some types of social gatherings, 'asked us to' and 'invited us to' are in fact synonymous. However, they still have slightly different connotations ('invited' is a bit more formal than 'asked'), and outside of those cases they do indeed have different meanings.
Argh I keep getting caught out on this one cos I use my regional phrasing: "The Swedish chef shouted us (to) lunch!" I am guessing that this is fairly regional or it would have been added by now? For me this would be the most idiomatic way to say this although treated works perfectly fine too. Not complaining about being marked wrong (or suggesting this option be included)... I am just interested to hear if others use this expression as well?!
de is only for plural expressions. For singular expressions, it's either den or det depending on the gender of the word, so since kock is a common gender noun, it must be den.
We normally use double (or triple, depending on how you see it) definiteness in expressions with adjective + determinate noun in Swedish, so this is just how our language works. In a few cases, when expressions are seen more as names, the article is left out. So when we speak about the Muppet show character Svenske kocken, that would be an example of that – in his case, 'Svenske kocken' is more of a name. The sentence above is not necessarily about him though, just about some swedish chef.