pronouns - “Both you and he are very happy” or "Both you and him are very happy"？

It's a sentence on a test paper:

Both you and he are very happy.

The sentence itself sounds awkward. Let's cast the awkwardness aside and look into the grammar. It's understandable that as part of the subject, "he" is the correct pronoun form. But I still wonder if "him" can be used anyway. We usually say "you and me" whatever the case, don't we?

Answer

Always split the sentence in such dilemma. Make those two people in concern separate and check what works!

You will go to London + Me I will go to London. SO, You and I will go to London.

She'll meet you + She will meet I me. SO, She'll meet you and me.

Take your sentence:

Both you and he are happy = You are happy + He is happy ~ Yes!

Both you and him are happy = You are happy + Him is happy ~ no, no!

And therefore, "We usually say 'You and me' in whatever cases as mentioned above."