Umm, the Neandertals never made it to North America. Or is that artistic license.
One of my favorite non-fiction books is The Neandertal Enigma by James Shreeve. He talks to a lot of the experts in the field–you get some interesting anecdotes (a Harvard grad student who spent his time knapping stone tools, usually from rocks he had hidden around Harvard grounds, and hurling stone-tipped spears in dead deer carcasses to examine the damage done to the spear-tips when hitting flesh and bone) and information on “modern humans” including the flowering of art in the Upper Paleolithic and some surprisingly old bone harpoons found in Africa. The writing is great–my copy has tons of lines and paragraphs underlined simply because I liked those particular passages. I highly recommend it.
Others have covered the first three questions pretty well, so I wanted to give some attention to this one.
It is possible that Neanderthal fossils could have inspired myths about giants, trolls, dwarves, elves, etc. I’m also familiar with some scholarly speculation that fossilized dinosaurs, mammoths, or other large animals could have inspired myths about dragons and other giant monsters. However, it seems unlikely to me that we could ever prove such a thing even if we could prove that ancient Scandanavians had uncovered Neanderthal remains.
We don’t know what the earliest forms of the Jötun myths were like. All we have is the versions that were eventually recorded. Since myths change over time, it’s hard to know if some resemblance between a mythical being and a real one means that the latter inspired the former (Neanderthal remains caused people to begin to believe in Jötun), the latter influenced the former (Neanderthal remains were taken as evidence of Jötun already believed to exist), or if the resemblance is entirely coincidental. That said, I’m not sure if pre-modern people would have thought that Neanderthal bones were anything unusual. Just a century ago many people believed that Neanderthal remains were really just the remains of deformed H. sapiens, and some creationists still believe this.
An interesting variant I’ve heard on this basic idea is that it wasn’t Neanderthal bones that inspired myths about fantastic creatures, but the Neanderthals themselves. Neanderthals and H. sapiens were contemporaries for a long time. We know that well before recorded history our distant ancestors told stories about plants, animals, and celestial bodies. Early H. sapiens were probably the same, so it doesn’t seem farfetched to assume that they had stories to about Neanderthals. But we can never know what these stories were like or even if they continued to be told for very long after the extinction of the Neanderthals, so we cannot know if they were the basis for any of the myths that would eventually be recorded thousands and thousands of years later.