A human that has one Neanderthal gene is not a Neanderthal. Therefore, Neanderthals are still extinct.
There's one other thing that I think people living in the Himalayas have, myself included, that helps and that is proportionately humongous lungs. This is probably evolution at work over the course of centuries.
That's funny. I had to Google it because I never heard about that before, but apparently it's extremely common!
According to Wikipedia this trait is common not only on asians but also native americans, so it predates Genghis Khan.
[1] http://www.amazon.com/Svante-P%C3%A4%C3%A4bo/e/B00GJ9XR7O/
"The Neanderthals Rediscovered", by Papagianni and Morse, has a good chronology of the species based on what is known to date.
Extinct? Hmmm. Get over it we are partly neanderthals, denisovans and so on.
Edit: Apologies to the downvoter, I should have substantiated that claim. I have added a source.
[1] http://www.infoplease.com/encyclopedia/society/neanderthal-m...