Biological classifications are one gigantic mess. There are multiple ways to define what qualifies as a "species". One of them is procreation and viable offspring, going by modern human DNA and the Neanderthal fragments contained in it we where one big happy family all along.
Depends on whether they were considered competing, and whether "we" were as organized, single-minded and competitive as this statement seems to imply - "we" probably weren't, not until larger kingdoms and empires started forming ~4000 years ago.
Ah yeah looks I wasn't on the mark here. The "outcompete" framing is more accurate for neanderthals but for many pleistocene extinctions "hunted to extinction" did happen in some cases so it was not a good comparison. Thanks!