Why lagar alone isn't acceptable here?
Lagar is almost never alone when referring to food. In swedish one does not just cook, one is either cooking some specific food, like lagar ett ägg, or cooking (food) in general, lagar mat
Is 'lagar' always followed by 'mat' when the specific food isn't stated?
As far as I know, yes that seems to be the case more often than not.
Maybe ask if he needs help with the cooking?
I wouldn't mind having a servant.
When is it medan vs innan?
Medan - while (during that time)
Innan - before
Is 'lagar' always followed by 'mat' when the type of food is unspecified?
I think (not very sure) that you can also say for example: "Jag lagar kyckling". I think is correct too.
Couldn't när work instead of medan?
När means when. Medan means while. Sure, they're close, just like in English, but not the same thing.