I’ll answer one part of the thread title, as far as I understand it… having nothing to do with Neanderthal DNA. How did Euros conquer the world?
This all mostly comes from a single book, called “Guns, Germs and Steel” by Jared Diamond. Here’s an executive summary.
In the battle between Euros and the “Indians” of the New World, Europeans decimated everyone they met and conquered numerous areas, precisely because of guns, germs, and steel. Natives in the New World couldn’t compete on even footing in a war where the enemy was blasting them with guns, stabbing them with swords, and giving them AIDS (just kidding).
The real question is how in the world did Euros get all these things natives didn’t.
In a word, agriculture.
Agriculture allowed nomadic groups to settle and build communities in areas that offered more stability than running around constantly searching for food. This allowed specialization. Different roles and occupations were created, in particular metallurgy and blacksmithing. Living in close quarters also offered the “opportunity” for new diseases to develop, including the black plague, which nearly wiped Euros out, but also offered a select few immunity to diseases that would prove fatal to Natives who were previously not exposed to what Euros were used to back home.
But why did Euros stumble across agriculture?
In a word, Mediterranean. More specifically the Fertile Crescent. The Fertile Crescent turned out to be the ideal geographic location for agriculture to develop far beyond anywhere else in the world.
Two charts here are helpful.
The chart below shows seed diversity amenable to agriculture at the pivotal time at which the New World was being conquered by Euros. The Fertile Crescent has the highest number of cultivable seeds by orders of magnitude compared to anywhere else in the world.
This next chart shows the diversity in mammals that could be domesticated for agriculture. Similar to seeds of fruits and vegetables, the sheer number of domesticable animals in this region wipes the floor with any other region in the world.
Because seed diversity and domesticable mammals are critically important to scalable agriculture, Europeans had thousands of years of a head start on the development of agriculture and, hence, modern weaponry, disease, and civilization overall.
Long story short, I wouldn’t really say the presence of Neanderthal, or any DNA besides that which offered resistance to communicable diseases in the Old World that the New World had not experienced, is what really resulted in that critical advantage for Euros. A key concept in genetics after all, is that environmental pressures shape DNA. It was mostly the fortune of having settled along the single place most amenable to agriculture in the entire world. Not the genetic lottery, more like the geographic lottery.