Lived the American dream
Bill Clinton was relating his late father's history. He wrote:

"...... After the war, Baron came to the United States, and, inspired by what he had learned from the soldier who called him “Little GI Joe,” he opened his own garage and started a family. He told me he had lived the American dream, with a thriving business and three children......"

1. does "lived the American dream" mean "lived in the way that he (Joe) had dreamed to live in one day in America"? Is this pheasing formal, casual or common? What's the ordinary expression for that?

2. if I replace the "live" with "live out", does the new text mean all the same? Or will the replacement break the cohesion or coherence of the original?

Question

Lived the American dream
Bill Clinton was relating his late father's history. He wrote:

"...... After the war, Baron came to the United States, and, inspired by what he had learned from the soldier who called him “Little GI Joe,” he opened his own garage and started a family. He told me he had lived the American dream, with a thriving business and three children......"

1. does "lived the American dream" mean "lived in the way that he (Joe) had dreamed to live in one day in America"? Is this pheasing formal, casual or common? What's the ordinary expression for that?

2. if I replace the "live" with "live out", does the new text mean all the same? Or will the replacement break the cohesion or coherence of the original?

Answer

"Living the American dream" means to become a part of American society, to be successful in business, and raise a family. It is an idealised life that people that moved to America hoped to have. Some did, some didn't.
It is a relatively common phrase, or at least it was, when it was an achievable dream  ;)
That's the only expression that covers the depth of meaning that that phrase has.
Adding out changes it a lot. When you live something out, it usually means you die at the end. So you typically wouldn't say that of yourself. "He lived out his final years in a run-down hospital".

Answer

"to live the American dream" means to have a good job, be rich, be happy all the time. It's idealized vision of American people.