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
"Laga" means "to fix". If you just say "laga" than it doesn't mean cook because you haven't specified that it's food you're fixing.
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.
Laga by itself means fix, so yes, you have to specify "mat" if you want to say "to cook".
Maybe ask if he needs help with the cooking?
When is it medan vs innan?
Medan - while (during that time)
Innan - before
I wouldn't mind having a servant.
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.
"Laga" by itself means "to fix". If you want to say "to cook" you have to specify that it's food you're fixing.
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.