Do individuals have any power over what they become or are their characters entirely determined by static factors such as family and race?
This is a large question. You will have to decide what side you are on. Basically, it deals with both "nature" versus "nurture" and "free will" versus " determinism." How much control or power does our environment have to shape our character, and how much of who we are are we born with?
Most people today will say that we are influenced by both nature (how we are when we are born) and nurture (what influences our environment has had).
Ask any mother, and she will say her children had definite personalities that they seemed to have from birth.This is like saying we have a "genetic predisposition" towards certain things, that some of our personality is "hard wired" into our genes. (Why is Johnny so happy all the time? He was born that way...)
On the other hand, much in our society is organized around the belief that environment can influence our personality and our actions. We care about pre-natal care because we don't want a child's options to be shut down before he or she is born (as is the case with severe fetal alcohol syndrome, for instance) and we read to our children because we think that that will make them smarter. Early psychologists actually believed that we could program a child to be anything we want her to be just by controlling her environment from the years 1 to 5.
This brings us to the question of free will, and whether or not we have any power to shape the person we will become. The most honest answer to this question is that we are both a product of our environment and a free agent with the power to decide for ourselves who and what we want to be.
Bilbo was not very adventurous by nature. As a hobbit, he was more excited about meal time than travel. Had Gandalf believed that people did not have the power to alter their nature, he might not have approached Bilbo. It is difficult to say whether or not Bilbo already possessed the courage and the character he developed over the course of the adventure. Perhaps it was always there within him; however, it might never have emerged had he not been in an environment that allowed it to grow.
In looking at the tag of the bottom of the question, the discussion is rather powerful. Tolkien would argue that individuals possess the capacity to choose their own sense of fate and destiny. These elements are dynamic and powerful. Individuals have the ability to rise above their own conditions and be better or worse than what might have been intended. No one would have ever considered a mere Hobbit to have changed the course of events in Middle Earth, but it was through the choices that Hobbits such as Bilbo had to make. Simply because he is deemed as a "hobbit" does not make him incapable of changing the course of events, altering the cosmic balance of good and evil. Tolkien would not say that this is easy. Rather, it is hard and challenging, as evidenced in Bilbo's. It is a challenge to overcome our own conditions whether they are familial, psychological, socio- economic, gender, racial/ethnic, or any other type of distinction that might prove to be challenging. It is a challenge, but one that Tolkien would say is within the reach of human beings to overcome. Tolkien might suggest that it is not only a struggle, but a responsibility of the caretakers of good to maintain its dominion over evil.