It is estimated up to 60% of the Neanderthal genome lives on in us. The ethnic European average for Neanderthal DNA is 2.7%. Their contributions to our immune systems are thought to have been particularly important. Having been in Europe and Asia much longer than Homo Sapiens, they had a lot more time to develop resistance/immunity to the germs/bacteria present there. Kids who had this genetic contribution from Neanderthals were more likely to survive.
Yup. They aren’t extinct, any more than all our other ancestors are extinct.