Self-reflection is the only way to realize our fears and mistakes, and correct them. Most people don't stop to reflect on their choices and actions, which in the long term causes anxiety.
I used to care about what people thought as well. As I get older, I see this tendency fade away to the point that I no longer care. I think it might be a matter of age as well. Teenagers like to explore through group formations. As people grow older, they each find their own individual meaning in life and usually get more independent. The gradual achievement of personal goals such as family or career, helps in self-confidence.
About successful people, I'm not sure. I'm assuming you mean successful in business or with their social circle. Some of these people are successful precisely because of their fears. A person who is dead scared of being alone, will create the biggest social circle possible. A person who is afraid of failure might spend the most productive and energetic years of his life trying to prove that he is worth something, becoming successful as a side effect.
That's the point though. In some cases we allow fears to get in our way and stop us doing things. If someone uses those fears in a positive way and actually does something then it is not really a problem.
That's true. But it can be a problem if you dedicate your life into something not because you are really enjoying it but because you want to prove to yourself (and others) that you are successful, strong, etc. There are many people I can think of who do things in order to impress and be accepted by others, not because they are really enjoying them. In this case they are not paralyzed by fear but they are using it as a wrong motivation.
Very true.