Rules of Grammar

OP, it doesn't sound stylishly correct to me for the simple reason that it is a repetition. I would say "the movie didn't interest either of us" in the second part of your sentence.

We had this all out before, at length, so I won't get back into the argument over whether it is right or wrong to say that either "My daughter and I" or "Me and my daughter" is wrong or right or whether there is a single context-free and black-and-white answer to the question at all.

Re #2 though, I am with Caballero on the looking for flights, and along similar lines, the more natural-sounding sentence for the African trip may be: "I am taking my daughter to Zimbabwe."

Back to OP question'
'I' is the Subject pronoun and 'me' is the Object pronoun, so that difference is correct (in English) for the OP example. Or was there a different possible problem?