Re: Do you have + article + singular or Do you have + any + singular?
Of the two choices that are you are "forced" to make, the one we all agree on is "do you have a pet" -- which I continue to find as unnatural as "Do you have a child?" in most situations.
I find "Have you read a good book lately" or "Have you seen a good movie lately" odd.
I find "Do you have an idea" or "Do you have a suggestion" far less likely than their plural counterparts as well, even though even one good idea or one good suggestion is all that is needed.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.