Question: Did Frank actually die when he cut the rope? Did they not find another way into the Galleria to check on him in case he only had a fractured leg, or did they just decide to leave him?
Answer: Frank fell several stories through the skylight of a mall. Even if he somehow survived the fall, Jack and Jason would have no way of rescuing him as the mall was buried under several meters of snow and the glass roof was precarious at best. Frank sacrificed himself so Jack could go to New York and rescue his son.
Answer: Yes. No, yes.