For the first sentence, the reason you have to make questions plural is because you are using the word HAVE. Have means multiple so it has to be questionS, not question. If you want to know if they have just one question, you would ask: "Do you have A question?" On the second sentence, the correct phrase is "I'm mad AT you". You cannot have a "feeling" be on top of someone, you can only direct it at them.

1. You can ask: Do you have a question? or Do you have any questions. Questions are plural in your example because it follows "any". 2. I agree with xarmanla

I believe that the "at" is used in the second sentence merely because certain prepositions go with certain words. As a counterexample, I could have a "crush on" someone and the "on" does not imply vertical positioning. I could also be "sweet on" someone. Additional, I could be "fed up with" a situation.