Does "how dare you" imply a question?
To say "how come you (didn't shut the fridge)?" or "how did you (forget to call)?" implies an obvious question.
So my question is: does "how dare you" mean why do you feel so emboldened to do so and so?
Solution 1:
On the one hand, the expression "how dare you?" is clearly a rhetorical question—that is, it isn't a question in search of an answer, like "how hot is it today?" but (in this case) an interrogatively framed expression of anger or outrage in response to an offensive action or statement by "you." In effect, the expression "how dare you [do or say X]?" amounts to saying "I find it [your action or statement X] unacceptable and intolerable."
On the other hand—perhaps in part because of its interrogative form—"how dare you?" issues a personalized verbal challenge that at some level invites a response: "How dare you?" hangs in the air in a way that "That's outrageous!" doesn't.
Most dictionaries of idioms don't mention "how dare you" as an idiomatic form. However, Cambridge Dictionary of American Idioms (2003) has this entry:
how dare you do sth esp. spoken I am very surprised and shocked by what you are doing | How dare you show up at my wedding? USAGE: usually shows that you think someone's behavior is very wrong: How dare he accuse of us of lying?
So my answer to the posted title of this question, is yes, "how dare you?" arguably does imply a question—a rhetorical one, at any rate—but it does not imply any doubt as to the speaker's verdict that the conduct or speech under discussion is utterly inappropriate.