A: No. She survives but loses an eye. She remains a major character for the rest of Season 1.
To understand the weight of "Under the Knife," we need context. The previous episodes saw Fish Mooney (Jada Pinkett Smith) apparently escaping Dr. Dulmacher’s (Colm Feore) horrific island—an "operating theater" for the rich where human organs are harvested. Meanwhile, Oswald Cobblepot (Robin Lord Taylor), aka The Penguin, has consolidated power by betraying everyone from his former mentor Fish to the Maroni crime family. Jim Gordon (Ben McKenzie) is a pariah in the GCPD, and Bruce Wayne (David Mazouz) is secretly investigating his parents’ murder with Selina Kyle (Camren Bicondova). Gotham - Season 1- Episode 20