Your sequence of events are wrong. If Humans and Neanderthals could produce viable offspring that were not mules then they did and some genetic material of Neanderthal is present in Human society today. It is that simple. It will take long term genetic study to prove one way or the other and no study to date is sufficient or even close to sufficient to determine this.
If they could not produce viable genetic offspring then there is none. This does not mean that Neanderthals and Humans had the occasional roll in the hay or even that there were long ternm stable relationships between the species.
Infanticide was common in premodern societies. Visibly deformed infants were almost always killed. It is an error to ignore that factor.
If they could not produce viable genetic offspring then there is none. This does not mean that Neanderthals and Humans had the occasional roll in the hay or even that there were long ternm stable relationships between the species.
No, it doesn't. Hell, it doesn't even mean that Humans and sperm whales didn't partake of the occasional roll in the hay or even long term stable relationships, but still, they probably didn't.