Additionally I'd say that 'How dare he' doesn't seem to follow the 'rule' because the [2nd] pronoun in 'I dare you' is the object of the verb, whereas the [only] pronoun in 'How dare he' is the subject of the verb. The expression is an abbreviation of 'How does he dare...?'. As 2006 says, there's an implied 'do/say that'; 'How dare he?' is a rhetorical question meaning 'How can he have the nerve/chutzpah/effrontery/cheek to behave in such a way?' or 'How can he be so bold/stupid/insensitive... [etc - any negative-loaded adjective to describe inappropriate behaviour]?'.
b
‘Dare’ is also called a ‘defective’ verb in some old grammar books because it can’t be used in all the moods and tenses.
"How dare he or him" is like an interrogative or exclamatory sentence. The subject is ‘he’. How he dare!? He dare (*dares) not do it; He dares (or dared) you to fight.
Most of us agree the nominative form of ‘he’ is considered correct.
And now the woman who screwed up the proximal verb on another thread will answer a grammar question...(drumroll please)
1) Yes, oughtn't she is the right tag, but...
2) I don't think I've ever heard this in AE speech. Here is a fascinating discussion of "ought" in American dialect. The author says that modern American speakers can't invert positive "ought" ("Ought I to go?") which is true; I don't think I've ever heard an American use this construction. He also says Americans can manage to invert the negative ("Oughtn't I to go?") but I very rarely hear anything like this. I would go on to add that although we use "ought" in the positive constantly, negative use is rare, certainly not "oughtn't" although I occasionally hear "ought not." A more common negative would be "you shouldn't go" or, as the above link discusses, "you shouldn't ought to go" which is incorrect grammatically.
Other AE speakers may have a different take on this.
"Right on the money" is indeed an idiom and it means "exactly right." Two links I googled said it derived from different sources, either horse-racing bets or the bull's eye in archery.