Well with the whole idea of the family hero, scapegoat, lost child and mascot, I think it depends. I do know one family that fits these roles perfectly, but then my family is completely different. I'm supposed to be the scapegoat/ lost child in my family, and I'm not. Nor is my older brother the family hero or my little brother the mascot. I think it depends on how the parents raise the family. My parents treated us as equals, so in a way we don't fit those roles. On the other hand, my friend's family put their three kids into those roles when they were younger, and they actually did turn out exactly as the birth order theory said they would.
In many cultures, and for most of human history, the answer would be yes. Birth order could affect everything from inheritance to economic position to legal status. There still are some parts of the world where the order of birth still matters, and even in some Western families this is still the practice. But in modern society, birth order doesn't hold nearly as much power as it once did.
In some cultures is does within the family, but at least here in North America a person's success is usallly in their own hands. I strongly believe we are in control of our own lives and fate and we decide if we will be successful or not by our efforts and determination.
Determine? No. Does it in general appear to have an impact, even today in the U.S., yes it does. They're still doing research on birth order and why a lot of successful people were oldest children.