"I can do the cooking" is correct. You would never say "I can do cooking". You would say "I can cook". I can cook = you are able to cook. You know how to do it. i.e., I can write, I can sing, I can draw, etc. I can do the cooking = you are offering to help. i.e., I can do the cooking when my parents come for dinner. I can do the cooking if you are too tired. etc.

They are equally correct. The answer to your question has nothing to do with the particular noun "cooking". Your underlying question is "when should I use the article 'the', and when should I omit it". The answer would be the same if the noun were "homework": "I can do (the) homework and everything else" Use "the" to specify that you have in mind some particular cooking. In your example, "the cooking" would mean the cooking that is done to take care of the mother. "The" makes a lot of sense in this context. Omit "the" to say that you do cooking in general. "I can do cooking" would mean something like "I am a competent cook". Neither one is THE right answer. The choice depends upon what you want to say.

To add to David Butler's correct comment, you are offering to help in all these examples: - I can do the cooking - I can do the cleaning - I can do the washing up - I can do the bathrooms - I can do the ironing. - I can do the weeding (garden) - I can do the editing/proofreading. - I can do the shopping. By the way it's a good idea to say "I can do the This if you don't mind doing the That". "I can do the driving if you don't mind navigating".