The possessive form, Emily's, is correct, and the word 'conclusive' seems a bit awkward to me. Perhaps 'suited' or 'appropriate for' would be more acceptable.
Emily is the possessive noun (Emily's). She possesses a scarf, although if she left her scarf in a public place it might not be hers much longer.
Emily braced for her exams
The primary thought of the sentence is that Emily (or your sister) has many friends. The name Emily and the tallest phrase are both "appositives" of your sister.
The victim in "A Rose for Emily" is Homer Barron. He becomes a victim of Emily Grierson's obsession and manipulation, leading to his mysterious disappearance and death. Emily's actions ultimately strip him of his agency and freedom.
Emily has a fascination with constellations.
Emily has many friends
After he was fired, the vengeful ex-employee gave company secrets to a competitor.
Emily used a sacred book for church.
C. Emily has many friends
Emily Post was an expert in the area of manners and decorum.
Emily said "Do not go in to the cave as it echoes emtily"
Emily has two sets of teeth.