Yes to both.
Batts was a member of the Gambino family. Paulie was a member of the Lucchese family. But since one of Paulie's people (Tommy) killed Batts, the Gambinos had the right to demand that Paulie order the murder of his own person to satisfy the Gambinos' need for vengeance.
You might wonder: Why wouldn't Paulie simply let the Gambinos take care of whacking Tommy? The answer is: Tommy was Paulie's responsibility. And, if Paulie let another family whack one of his own people, it would set a bad precedent. Paulie would lose face, then lose power.