Bob is a US citizen married to Lily a non-US resident. They live together in Europe with their US citizen daughter (born abroad). If Bob was to pass away, would his wife Lily be eligible to live in the US with their minor daughter?
If Bob were to pass away, Lily can self-petition to immigrate to the US as a widow(er) of a US citizen within 2 years of Bob's death. (If Bob had filed a petition for Lily before his death, but she had not immigrated yet, it would automatically convert to a widow(er) petition; if no petition had been filed yet, then she has to file a new petition within 2 years of his death.)
If she misses the 2 years, then she can no longer has any path to immigrate to the US through Bob; she could see if her daughter will petition her when the daughter turns 21.
No. He would have to sponsor her green card before passing away. He can do this now even if they are not currently residing in the U.S. Otherwise she will not have any special entitlement to reside in the U.S. with their daughter until the daughter is able to sponsor the parent upon reaching the age of 21.