I don't understand this. Assuming no inbreeding, families that consistently have 2 kids roughly double in size every generation - a 600-year-old family would have 1 billion unique descendants by now. Of course, there is a ton of inbreeding - in the technical sense - but I still have a hard time imagining a 600 year continuous family line that isn't a pretty arbitrary chart of certain descendants here and there. It's very probable that some of them became poor too!
It's a patrilineal family line. Women of the family who bear children are (in most of Europe) considered to have exited the family. In addition, sufficiently distant relationships often break off into cadet branches and stuff like that. There's a whole field behind this weird stuff.
That's not correct; once you reach out a few generations the branch becomes a web as distant relations marry. That's the only way it can work if you think about it.
Your maths seems off. 600 years at, say, 25 years per generation, yields 24 generations. Doubling each generation gives a factor of 2^24, or around 16 million, not 1 billion.
It's not forgetting death that's the error; it's forgetting that people have 2^30 ancestors from 30 generations ago rather than just one. Everyone's family tree grows exponentially, but they all overlap.
There had been roughly 120 billion humanoids living (and dying) so far. Not sure where cut-off line between apes and homo sapiens exactly is though. The growth is +- exponential. So 1 billion doesn't mean much (and yes all overlaps with other trees, so voila we have present). Plus many people died before having kids, tough times it were.
A family having two kids would typically have one girl and one boy, which means the name ("the family") would carry on only half of the time, which means there would be no growth.
You'd need a higher rate to grow just based on that, plus there are untimely deaths and non-reproducing offspring...
