With Sentences 3, 4, 5 and 6, the verb is a be-verb, meaning that the pronoun is a complement, rather than a subject or an object. If the verb were “play,” then “her” would be an object and therefore correct. I would tend to agree that complements should also be objective (her/him/them) in most cases.
I think that it becomes tricky in sentence structures such as, “It was I who…” because, although “I” is being used as a grammatical complement, “It” is being used merely as a dummy subject, while the real subject is clearly “I.”
Perhaps this is why we say, “This is she” or “I am he” on the telephone: because the true subject is the name to which the pronoun points.