mad at

1a. She is mad at her brother.
1b. She is mad at her brother’s action.
Are both sentences correct?

They’re okay, though ‘mad with’ would be more usual in sentence 1.

1a. She is excited with/at her brother.
1b. She is excited at her brother’s action.
Are these sentences OK?

1 doesn’t make logical sense. Why would she be excited at or with her brother?

Just to let you know, I think in the US “mad with her brother” would be very odd and unusual. The common expression here is “mad at her brother”, though we do say “angry with her brother”. I wonder why we differentiate between these two words - I never thought about it before.

Thanks for the intervention, Luschen.
I think I was confusing ‘mad at’ and ‘angry with’ and you are right - ‘mad with’ would be odd in the UK too.
(Though I consider ‘mad at’ to be a very ‘Americanized’ expression anyway, and not used so widely in the UK, where ‘cross’, ‘annoyed’ or ‘angry’ tend to be used instead.)

[color=blue]I would say ‘(angry(mad) at’, though I am aware of ‘angry with’ being used. On the other hand, I am only aware of ‘happy with’. Does any one say 'happy at my brother? I think not.