Chromosomes cross over. The analogy falls apart, because by that time there are no identical chromosomes (except the Y chromosome, for reasons that should be obvious, but that’s only true in direct male descent).
Yes, it’s quite common. The chiasmata formed by crossing over are an important factor in keeping homologous pairs correctly bioriented during meiosis I (it happens only during meiosis, because homologues don’t pair up during mitosis). There are systems to ensure correct segregation of nonrecombinant chromosomes, but generally speaking, most chromosome pairs have at least one crossover point during meiosis.
Yes and no. As mentioned, during egg/sperm production, the chromosomes cross over and swap pieces. So genes do get shuffled around. Though there are probably some genes that are right next to each other and tend to stick together fairly strongly (i.e. the chromosome rarely crosses over right at that point), so the assumption of perfect independence is probably wrong, too. But over that many generations, close enough – there are going to be much much bigger sources of error in the estimate.
Also, the estimate I did above doesn’t depend on the number of generations, so the best estimate is still about a 50/50 chance ERII got one of WtCs. Though I’m pretty sure I missed a factor of two one or more places in my math, and the estimates are pretty WAG.
With the decimation of the Amerindians, it’s pretty easy to understand how all those who remain today almost certainly have European ancestors, but I don’t know that enough time has passed such we can assume gene flow back to the rest of world in such a way that all non-American populations have American ancestors.
Still, if the identical ancestor point is < 13k years ago, then it must be true since clearly the extant Amerindian populations have Amerindian ancestors from that timeframe (or call them whatever population was living in the Americas at that time).