Old English or Anglo-Saxon is a separate language, spoken vaguely around AD 500-1000. Middle English is the next half-millennium, and then Modern English is the period since then. "Have you the..." is contemporary Modern English that was fully standard until the mid 20th century, and is a less common but still current variant in use to this day.
So no, not Old English.
"you have the salt?" would only be understood as a question with the proper rising inflection at the end of the sentence. Although your translation would be understood, duolingo expects you to use the auxiliary construction (DO you have the salt?) to translate questions.
Why is it that way in Swedish? Because Swedish expresses questions like that.
Word for word, it is "Have you salt-the?".
In English this is traditionally "Have you the salt?" (i.e. the question is formed exactly as in Swedish), but the modern English tendency is towards auxiliary verbs, so we tend to modify it to "Have you got the salt?" or "Do you have the salt?".
It's the English that is a bit baffling. The Swedish is rather straight-forward.
Hmm. To me the first answer is "do you have the salt" then "have you the salt" (old fashioned). To me "have you got the salt" means something more active than mere possession, along the lines of "have you acquired/brought/taken the salt.". For example, someone is asked to bring things for the supper table, they come and one asks "have you got the salt". So, the question:. How broad is the meaning of "har"?