According to the traditional rules of English grammar, sentence 1 is wrong and sentence 2 is correct. The case of the two pronouns differs: she/I and her/me are in the same case, but her/I are in different cases.
Nevertheless, you will hear many, perhaps most, native speakers in the US use sentence 1. I still recommend using sentence 2 because at least some will judge sentence 1 to be poor English and because everyone will understand and be comfortable with sentence 2.
In the case of sentences 3 and 4, it is recommended to put the first person last in a list of pronouns. So sentence 3 is what is recommended. That is not, in my opinion, a rule of English grammar, but rather it is considered good manners to mention others before mentioning yourself.
EDIT: Case is just vestigial in English, and so many mistakes of case are not noticed by most native speakers, even by quite a few educated native speakers. Thus, worrying about case is probably not the most important aspect about learning English. If your native language is a Romance or Germanic language, the traditional rules for the case of pronouns in English are quite similar to those in your native language.
EDIT 2: No article is grammatically needed before "headphones." It might be clearer to say "a pair of headphones" because the given structure does not distinguish between a single pair and a truckload of pairs: either is more than one headphone. This is being fussy, however, because it usually would be clear in context.