The first sentence is fine. The second sentence should be either 'I think he's got the flu' or 'I think he's got flu'. You can use the definite article, 'the flu', or (in GB English, at least) you can omit it and just say 'flu'. I would usually omit the article as this is the commoner form in British English. Note that it's very rare to use the full form of 'has' before 'got' unless you are emphasising or contradicting a statement. Otherwise, we say 'he's got'. Poor little chap. Give him plenty of drinks and cuddles.

A native speaker would say: He's not feeling well, I think he has the flu. "Unwell" is a bit formal for speech but you can say it. "Not feeling well" is more is more common. Some people erroneously substitute "good" for well. This is not correct. The question is about health, not character. "A" flu? No. Note: my comments are for American English only since I'm an American.

Yes, it sounds natural. I might be more likely to say "the" flu, rather than "a" flu, but "a" flu still sounds natural.

I agree that "a flu" sounds a bit strange, "the flu" is much more common. You could say that a person caught "a" stomach bug, or "a" disease.

I guess that: " He's not feeling well , I think he's got flu" sounds more natural