Ya shudda come to me first, I cudda helped you.
The picture you posted above, for example. The cat was a stray that was on the set. The crew began feeding it and Brando used it to give him a touch of tenderness.
Luca Brasi (Lenny Montana) below. He was an ex professional wrestler watching them shoot on location. Coppola thought he was one scary looking guy and put him in the movie. In the scene pictured below was accomplished at Montana's suggestion. He is doing a training exercise he learned as a wrestler. The part of him practicing his lines? That was real, he was not a pro and was nervous.
There are literally dozen's of other little things like those mentioned here that added up to the two best movies ever, IMHO.