There are two separate issues here: grammatical correctness and common usage. All of your examples are grammatically correct but common usage requires use of an article. It is important to understand how an article changes the meaning. Once you understand the meaning of "I have backache", you may understand better why we do not say that, even though it is not wrong. Without an article, "backache" is the name of a condition, or disease like "influenza". Without an article, it does not describe your particular backache (which is what you normally want to describe), but rather all backaches. We say "I have influenza" to name the disease we have. We say "I have a backache" to talk about your particular backache. Your backache is personal. Ouch! It needs an article. If you said "I have backache" you would be turning "backache" into a disease. That is not wrong. We just don't think of it that way. I can't say why; it's just customary.

Speaking as an American, I always use the article for pains and aches. For example, I have a backache, I have a headache, I have a stomach ache. But if you use “pain” you wouldn’t use the article: I have back pain. I have leg pain. But in these cases I would say it’s more common to hear someone say: My back hurts. My leg hurts. You would also use the article in the following: I have a cold. I have the flu.

I agree with David. It sounds unnatural to me, especially for pains (headache, toothache etc)...I would always use "a". But maybe this works colloquially for a specific native English country. For major illnesses though, many don't use articles: I've got diabetes ✅, I've got cancer✅, I've got vertigo✅, I've got amnesia✅ .................................................................. I've got flu/cold❌ >>> I've got the flu/ a cold✅ I've got a fever✅ I've got a diarrhea✅

Apparently this is a usage that differs by region or dialect. I have never heard these words used without the article. I would suggest that you use the article with all of them in your own speaking or writing, but don’t be surprised if you hear someone use them without it. I’m a native speaker of American English. Perhaps our British friends can tell us whether omitting the article in these cases is British usage.