The question is fine.
The response is not. The point about the question is that it is not really a question demanding an answer. It is something said as a sign of indignation.
With the quotation marks, you have to quote what Tom said: Tom said to her, "How dare you touch my car?"
As reported speech, write: Tom asked her how she dared to touch his car.
Neither of these is particularly natural.
I heard someone say it years ago.
I don't see that How dare you say such a thing to me!, when said in anger, is a question at all, and I wouldn't want to transcribe it with a question mark. And since it's not a question, it doesn't elicit an answer.