the correct grammar is to always mention yourself last
That was exactly why I mentioned this case. What exactly separates the grammar of “I (and Dave) am going” vs “Dave (and I) is going”? They’re both “subject plus verb”, indicating that the order of the coordinations is a matter of style, not grammar.
As to the subjective vs objective cases, the reasons I brought that up is that grammar seeks to explain the rules of language as native speakers use it. Therefore, the only “wrong” grammar is constructions that natives wouldn’t use, such as (eg) mimicking a Japanese grammar structure and saying “I the house to go.” Since native speakers often use cases like “Me and Dave are going spelunking”, it’s not wrong, merely a different register of speaking.
Accepting only prestige dialects as “correct” grammar is why people continue to think things like “black people don’t speak right”, despite the fact that African American English has the same consistency of grammar that the prestige dialect does.