Whoops! Accidentally wrote 'cheese wine' instead of 'choose wine'...
Must have been thinking too much about cheese...
In that case, I'd say that the literal English translation, ‘choose wine’, is wrong; it should be ‘choose the wine’.
"Hättest du" does not equal "hattest du". The former is conditional, which translates to "ville du have", and the latter is past tense, "havde du".
No 'at'? I thought that was the point of these infinitive lessons: at drikke; at vælge?
This is the "naked" infinitive, just the verb without at. It's always used with modal verbs (kunne, ville, skulle, turde, and so on). In that regard it functions just like in English. In English you also use the naked infinitive (without 'to') after modal verbs, like "can", "will", "should", or "dare", for instance.
"Will you choose the wine today?"