Yes, the Mati airport (Old name) and the (New name) "Francisco
Bangoy" International airport in Davao city, Philippines are the
same airport.
NO. Mati Airport is, as the name suggests, IN Mati City. That's
in Davao Oriental. While Francisco Bangoy International Airport is
in Davao City in Davao Del Sur.
Francisco Bangoy International Airport is NOT a new name. It is
the first name of an entriely new airport in Davao City.