yes, we do. Modern humans are a few % neanderathal. We also know that due to missing sections of DNA that some of the Neanderthal traits came at a survival disadvantage in the Hybrid.
Now given a few more 10's of K or 100's of K years of separation, we may have had species incapable of interbreeding.
No, we do not.
As I was saying, there exists a considerable possibility that those genes did not enter the human gene pool through interbreding with Neanderthals, but through interbreeding with an intermediate population. The Skhul/Qafzeh hominids.
Consider also that Neanderthal genes seem to have entered the human genepool early, possibly before we overlapped in range with them. Also, after a few breeding events at this time, there seems to have been very little further interbreeding over all the milennia when we did overlap with them.
The scenario as I understand it could be: Humans in Africa, Neanderthals in the Near East, Denisovans in Europe and North Asia, The Skhul/Qafzeh hominids in the middle east.
Neanderthals exapnd their range, pushing the hybrid species known as Denisovans out of Europe, and interbreeding with the Skhul/Qafzehs, who are intermediate them and humans.
Humans push out of Africa, encountering the the Skhul/Qafzeh and with some difficulty interbreeding with them, absorbing Neanderthal genes from them. We continue expanding and the Skhul/Qafzeh disappear like all the rest will. We then overlap with the Neanderthals for a considerable amount of time but are too different to interbreed.
Classic
ring species.
Of course this is just one scenario, but it does demonstrate that there are scenarios where Neanderthal genes entered the human population without direct interbreeding.
We've also known abut the fitness costs for some time, due to the areas of the human genome that lack neanderthal genes completly, the genes involved in reproduction.