This is my first time reading it, but the movie is one of my favorites. I've heard that it takes a few readings to get ahold of Kesey's style, so I'm naturally a little confused.
Why the hell did Henry's dad take all of the family's money and leave? All I can gather is that the strain of frontier living was too much?

Why did Myra marry Henry, when the book made great pains to describe how awkward and pathetic he seemed in comparison to her other suitors? Was it simply rebellion against her family/lifestyle? Leland describes her as a beatnik....

What did Joe Ben's dad do to instill such fear in his son? Was it just the fact that he was a lothario and Joby wanted to be a settled man?

Any relation to the Paul Newman movie

Yes. It was written about a decade before the movie came out. Ironically enough, it mentions a Paul Newman movie at one point.

Politics aside, he is my favorite actor of all time. He had some great roles.