Yes, correct, in informal (반말) style.

I may be wrong, but I think it should be "우리는 진짜로 행복해.”

Correct

It's just my opinion. "우리는 정말(로) 행복해." Almost similar expression. Depending on the situation or the asker's intention, the reply could be changed, I guess. So you could say in korean, " 우리는 정말 행복해"