What is correct my brothers and me or my brothers and I?

Is it grammatically correct to say me and my friend?

Yes, it's grammatical (at least in terms of native spoken English) to say 'me and my friends' even in nominative position, but it's a disfavored construction because most people regard it as sounding uneducated.

Is it grammatically correct to say me and my?

It is correct to say, "my friends did something with me", but not "me did something with my friends". While "me and my friends did something" is perfectly understandable, it's grammatically wrong.

What is the proper way to say me and my friends?

“My friends and I” is grammatically correct when it is placed before the verb and is the subject of the sentence. “I and my friends” is also correct but less common as it is more polite to put your friends first.

Is it grammatically correct to say me and someone?

I and someone is grammatical; me and someone is not strictly grammatical, but is very common; I and someone we is not grammatical, and sounds wrong to native English speakers.