To some degree. Fiber supplementation has been shown to reduce the likelihood of episodes of diverticulitis, but not completely prevent them.
It could. High fiber diet is one of the best ways to prevent diverticular disease. You also need to have adequate fluids as well, so that the fiber will form a bulky stool that is soft and easy to pass, thereby minimizing constipation, and straining.
Maybe. But if you eat high fiber and drink plenty of fluids, avoid constipation, this will reduce the likelihood of diverticulitis. If you have diverticulosis without infection it will not go away but the above rec's will help you to remain symptom free.