Sanna, the way you're trying to explain it just isn't at all how English grammar works. The thing that matters is if you take away the other person, would you say "me" or would you say "I"?
Gaia and (me/I) are having afternoon tea.
Me am having afternoon tea.
I am having afternoon tea.
Gaia and I are having afternoon tea.
Would you like to join Gaia and (I/me)?
Would you like to join I?
Would you like to join me?
Would you like to join Gaia and me?
The only funny case is the predicate nominative, which is where you use subject pronouns (I/thou/they/we/you/they) instead of object pronouns (I/thee/them/us/ye/them) after the verb "to be" because "to be" is basically an equal sign, and the sentence is reversible.
The people having tea are Gaia and I.
Gaia and I are the people having tea.
That's not something that applies in OP's issue though.
Anyway, what I explained in my earlier post about prescriptive vs descriptive grammar applies and this whole thread is a non-issue.