You can find all this, and much more detail, in Harlan Lebo's excellent "The Godfather Legacy." To answer your questions briefly:
1. FFC was a young, relatively unknown director at the time. Paramount hired him mainly because they thought he could do "The Godfather" on the cheap(!).
2. Brando was generally cooperative on the set and was a calming influence. Brando didn't have to beg for the part. Mario Puzo wanted him for Don Corleone. So did FFC. He rejected the script initially because he'd never before played an Italian. But he thought about it. Paramount execs didn't want Brando at all. But FFC showed up at Brando's home with a hand-held camera and did some screen tests with him. When Paramount execs saw the tests, they enthusiastically endorsed Brando.
There are plenty of other great stories in Lebo's book.