Interesting question! I'm unsure how to answer it, but I'll tell what I've found out.
Don't go cooking for him every time he pays us a visit.
I've looked up Swan's "Practical English Usage", and its Unit 228 says that we use go with an -ing verb to describe activities with the following characteristics:
- People move about (dancing, climbing, skating).
"Cooking" seems almost to fit here - one could move about while cooking, although not to such an extent as in skating, maybe with very rare exceptions. You could object: "but one who is cooking is anchored to his appliances, to the table, the stove". But a skater could also be anchored to the skating rink and thus have his freedom to move about curtailed.
- There's no fixed beginning or end point to the activity.
This seems also almost to fit with "cooking" - especially in a bare sentence like "Oh, hang it all, I'll go cooking" (I will cook, and cook, and cook..). On the other hand, one cannot cook a single dish as long as one wishes. It has to become ready or spoiled at some particular moment.
Swan provides an example of a sentence where the "go .. ing" structure is used wrongly:
- I'll go boxing. (Since a boxing match is an activity that has a definite beginning and end, the sentence is wrong, according to Swan).
And, when we add the next parts of your sentence, "for him" and "every time he pays us a visit", this seems to make the activity more definite in time coordinates. Each time he pays a visit, you'd want to start cooking. You'll likely finish when he leaves.
In the end, I'm unsure. Let's wait for what native English speakers will say. Maybe this "go cooking" gives additional emphasis to the sentence, implying that the person is ready to spend indefinite amounts of time cooking for the visitor, so dear the visitor is to him/her.
Or maybe the verb "cooking" by definition implies activities having a beginning and an end (when the dish has been cooked), and thus looks outlandish to the native speaker.
Maybe a native speaker will even read "I'll go cooking" as "I'll become cooked"?
- Just to cap it all, Swan also says that the third kind of situation where "go .. ing" structure is used is in talking about looking for or collecting things: "I think I'll go shopping this afternoon."
I would love to find the pages in Quirk et al. or Huddleston and Pullum where these constructions are described.