You "have the flu" or "have a case of the flu." I have never heard anyone say they "have a flu" unless they are talking about their chimney.

"Have the flu." However, there are some alternatives here: I have a nasty flu. I have a bad case of the flu. He has a mild flu. But never: I have a flu. I don't know why!

Have a cold, have the flu. A cold is non-specific, "The Flu" is derived from: "Influenza". You can't have A flu, because there's only one type.