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I've studied genetics a bit, certainly enough to address your question.

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Thus even if you could be certain you were descended from a particular person back that far, you might not actually possess any of his genes.

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Humans have a high degree of genetic conservation, that is they share many genes with each other. Depending on what source you like, we're something like 98.5 to 99 percent genetically identical to chimps. That is unsurprising given our estimates that we have a common ancestor with chimps 15 million years ago, NOT to be confused with having descended from chimps. The greater the proportion of any particular regional group of humans you have, the more likely you are to have the same genes. The number 100 generations is arbitrary. Every gene has a number of forms, called alleles. Different gene loci have different number of possible alleles so loci with only a few alleles might be very conserved (read: they don't change) compared to your ancestors. Race, ethnicity, etc. are all very arbitrary concepts. The best you can do and remain scientific is to talk about regional, shared ancestry.

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I was not talking about overall genetic conservation, but about particular conservation. You get exactly equal numbers of genes from your parents. But of the genes you get from one, they are not necessarily an equal mix of those that parent received from your grandparents. This is of course why certain traits including facial likenesses show up in following generations. Thus when looking at desecendent generations, a certain anscestor's genetic contribution can be getting proportionately smaller than only halving it each generation would make it.

Actually I was careless in just saying 100 generations when it can easily be determined that after 16 generations there is a possibility that your descendants no longer possess your genetic traits provided that you are not duplicated in their family tree. We simply don't possess enough genes to inherit all the contributions of previous ancestors, and not very far back either. As I said, this has nothing to do with overall genetic conservation of the species in the evolutionary timeline.