A sentence becomes negative by specific words. Most of those have 'no' in it. For example
No
Nobody
No one
Never
Not
Nowhere
...
And yes a sentence without a subject can be negative. Here's an example:
Stop the car! -- positive
Do not stop the car! -- negative
Do your homework!
Don't do drugs!
I believe I understand your concern because similar confusion arose at some point in my English study, too, due to a linguistic difference between my native tongue and English. Your language is perhaps like mine, Thai, where you answer to the truth condition of the statement of the question rather than the truth itself. For example, if you ask, in Thai, "aren't you going to school today?" and the answer is "yes," the answerer means they are not going to school. "Yes" here is "yes, (it is true that I am not going to school)," not "yes, (I'm going to school.)"
I think it helps if you think of English as somewhat simpler than that. In English, whether you answer "yes" or "no" only depends on the reality and has nothing to do with how the question is worded. Whether the question is "do you know?" or "don't you know?," "yes" and "no" still have the same meanings: in both cases, if you answer "yes," it means "you know," and if you answer "no," it means "you don't know."
Answering with plain "yes" or "no," however, may sound too flat and could sometimes be misunderstood, so it's safer to respond with a complete answer like "yes, I do." (meaning, you know) or "no, I don't" (meaning, you don't know) if you can't think of anything fancier like "no, I never knew," or "no, this is the first time I've heard about it!"
As stated by other Answerers, one thing to consider is that sometimes "don't you know?" is a rhetorical question, meaning the speaker doesn't actually expect an answer, so you don't have to answer them. You usually can guess from context whether it is an actual question or not.
Best Answer
As @blubberguy says, "any" is not a negative word. Negative words are words like "no", "not", "never", and "none".
Like many "grammar rules" that you hear, "never use a double negative" is not really a valid rule. It is a guideline.
Yes, double negatives are sometimes just bad grammar. "I am not never gonna help Bob." What does that mean? Taken literally, it would seem to say that you ARE going to help Bob: "not" would mean to reverse "never".
But there are many cases where a double negative is perfectly valid grammatically. Sometimes we use a double negative for emphasis. "I am not, no way, ever going to help Bob." "Not" and "no" are both negatives, but they are clearly not reversing each other. They are repetition for emphasis.
Sometimes we want to reverse a negative. Like: "I am not refusing to help Bob." In this sentence, "not" and "refusing" are both negative words. But people often say things like this as a milder form of a positive. The person doesn't say he is "agreeing to help Bob", just that he is "not refusing". He may be helping Bob reluctantly, or he hasn't decided yet whether to agree or refuse. So he is not refusing, but he isn't necessarily actually agreeing either.
The problem with double negatives used this way is that it can be confusing. I recall a discussion of this that I read years ago where the writer mentioned a sentence he read in a newspaper about some political development that had a string of negatives. I forget the exact example but it was something like, "Congress decided not to refuse to delay the vote on repeal of the law banning X." This sentence has a number of words that in context are negative: "not", "refuse", "delay", "repeal", "banning". So was this action by Congress for or against X? You have to study it and think it through.
You often hear people say, "two negatives make a positive". Apparently they are trying to apply the rules of arithmetic to grammar, and are thinking of an expression like -(-3)=3. But who says the rules of arithmetic apply to grammar? And even if they do, sure, multiplying two negatives gives a positive, but adding two negatives gives a bigger negative.