Questions & Answers about Meine Schwester und ich lesen abwechselnd vor.
Because meine Schwester is the subject of the sentence, so it is in the nominative case.
Here, my sister and I are the people doing the action, so German uses the nominative: Meine Schwester und ich.
For the same reason: ich is nominative, and this sentence needs a subject pronoun.
- ich = I
- mich = me
In English, people often say me and my sister in casual speech, but in German the standard form here is meine Schwester und ich, because both parts are subjects.