The sentence is correct grammatically, but it's unnatural and could be incorrect culturally. 1) If you are talking to your friend who looks upset, Like miho-san mentioned above, ”怒ってる？” would be natural. 怒っていますか？is also possible if you want to make it a bit politer. 2) If you are talking to someone superior(teacher, boss, elderies, etc.) who looks upset, maybe it's better not to ask whether he/she is upset or not. Just apologize if you have done something wrong. It's not only about this sentence. In general, it's considered impolite to ask superiors' emotion directly like "Are you glad, boss?" Instead, you may say "I guess it should have been fun for you." or something like that.

This sentence is right, but it sounds a little straightforwardly.　 There are few opportunities for using this phrase in daily life, I think. because it's premised on a relationship. You can also say to your friends or people close to you ”（わたしのこと）怒ってる？". "もしかして、怒ってる？" could be better. （もしかして = I'm wondering / I'm not sure )

確かに　これで意味は理解できます ただし　相手が怒っていると感じた場合、自分の立場を下げて（謙遜して）相手に尋ねるのが普通です 「私を怒られていますでしょう」　というのが通例です もし、本当に怒ってたとすれば、事態を悪くする可能性があります。 文法論ではないのですが、参考までに加筆します

でも　通常は　怒られていますよね？と聞くこと多いです

完全に伝わります。