Asked ''Who ate the cake?'', many people might answer ''Me and him.'', and look upon ''He and I.'' as pretentious.
These same people, however, would ridicule you for saying ''Me ate the cake.''.
Thus we have some evidence of a ''folk grammar'', if you will, with no small intolerance of infractions.
My question is, to what extent are such grammars similar across languages, geographical regions, and historical times? Has anyone ever described systematics in them? Has anyone ever theorized about their origin?
I'll take a different tack from Prof. Fidelholtz. The other issue you raise involves vernacular dialects and attitudes towards them and towards Standard English. One of the unfortunate features of Standard English as an ideology is its prescriptivism, that is, the belief that there is a right way to use English and to deviate from that shows one's ignorance, lack of culture, inadequacy of training, and perhaps even moral turpitude. The reality is that languages have dialects, and often one of those dialects, because of accidents of political and economic power, becomes Standard, and its speakers then set themselves above those who don't speak it.
But non-standard dialects differ from standard only in social prestige and power, not in flexibility, expressiveness, or complexity of structure. They don't lack grammar; they have different grammars. Your example of "me and him" as subject, vs. "me", is a good example. In both colloquial and even informal spoken standard English, we often use "me" or "her" where Standard English rules might seem to require "I" or "she". In answer to the question, "Who's there?", we might respond, "Me", but not "I", or "Her", but not "she". We tend to use objective pronouns when the pronoun carries new information, even in subject position. If the subject is just a single pronoun, it's usually not carrying new information, but if it's conjoined, as in "Me and her went to a movie", then there is more information, both pronouns get stressed, and it feels more natural than "She and I went to a movie."
The fact that these objective pronouns are commonly used in a lot of unexpected ways goes back quite a ways. HLMencken, in his The American Language, provides a very interesting treatment of it from about 70 years ago, and it's clearly a lot older than that.
It's just not accepted as Standard English.