"Cook lunch" would be more common, but "cook a lunch" is perfectly fine.

If you're talking specifically about the mid-day meal, you can say 'cook the lunch' (i.e.: 'It's my turn for cooking the luch today'). But for talking about a meal in general use 'cook a meal' or 'cook the meal' (i.e.: 'I have to cook a meal at least once every day', 'Yesterday I cooked a delicious meal for dinner').

'A lunch' is only used when modifying 'lunch'. Ex. I cooked a healthy lunch. I cooked a lunch consisting of... I cooked a lunch that... 'A lunch' on its own IS acceptable, but if you don't intend to modify 'lunch' then it's more natural to drop the article.

"Cook a lunch" may be correct in situations where one is cooking a meal for sometime in the future. For instance, one might "cook a lunch" in the evening to bring to work the next day, although you could also just "cook lunch for tomorrow." It may not be formally correct but I definitely hear it used colloquially. However, most of the time "cook lunch" is used.

No, 'cook a lunch" is never perfectly fine, "cook lunch" would be the correct way of saying it if it is a regular thing, like "I'm going to cook lunch" or "I cook lunch every day". Otherwise "Cook a meal", like below.