It has been a while since I read the book, but Iwo Jima was the last outlying bastion the Japanese had before the "Home Islands" and it was very important that they hold it. It is spelled out by James Bradley, but I can't remember the details. I do remember the details of what the Japanese did to the poor American Medic they caprured and tortured for 3 days. Sickening, to say the least.
Not actually true. Iwo had little to no strategic importance. You may be thinking of Okinawa, which was of enormous strategic value.
The American military originally wanted Iwo simply as a potential emergency landing field for American bombers. Highly unlikely that it was worth the price paid. Better to bypass it and let the Japs starve.