[Me and my twin brother] are having a birthday party
Sorry to disagree, but "me and my twin brother" is non standard English.
I is a subject pronoun, and
me is an object pronoun.
An easy way to check if you have used the correct pronoun is to drop the "and twin brother" and see if the resulting sentence is grammatically correct.
Me
and my twin brother am having a birthday party [incorrect]
My twin brother and I
am having a birthday party [correct]