I was also taught from grade school through college that "I" always comes second, but "me" always comes first. In a nutshell "XXXXXXX and me" is never correct. I majored in English and have worked as a salaried writer for almost 30 years, and I can't believe how many resources there are online espousing use of "and me." (Have I discovered "Fake Grammar"?)
If you think about what any "me and" statement is communicating, it makes sense to state "me" first. For example:
"I hope you'll join me and the leadership team at our annual meeting."
Stating "me" first makes the statement sound sincere. You'd never say, "I hope you'll join the team in going to that thing I won't be going to." Why would you hope someone would do something with someone else that you're not involved in? You wouldn't, so saying you would have some hope about it is insincere. You're only going to hope someone attends something that you will also be attending, ostensibly because you hope to see that person at the event. The other people being there are beside the point. You can only express your own "hope," which is that "you'll join me."