I admit my sort of base assumption is that simple quantitative metrics such as n% African/n% Neanderthal peddled by the likes of online genetics testing labs are grossly oversimplified at best and probably borderline meaningless at worst...
Regarding ancestry testing: it’s pretty valid, in general. DNA extracted from ancient human remains has completely revolutionized our understanding of recent, ancient, and extremely ancient human history. The online testing companies use the same methods and data sets. The way they present results boils a lot of different stuff down to a single number, so nuance and some fidelity is inevitably lost, but that’s always an issue when you try and make a nuanced subject understandable to laypeople. But, generally speaking, if an online test says that your DNA is 5% Neanderthal whereas someone else’s is 1%, or that you’re 20% Western European and 40% Eastern European, that is probably a valid way to present things.
Species is a squishy concept. You might be amazed at how much intense arguing that continues to be about what delineates two closely related species. I think it's hard to consider Neanderthals a separate species, given what we now know thanks to ancient DNA. I think the term sub-species makes more sense.
> What does it mean to have n٪ Neanderthal DNA if that DNA is found in a significant proportion of "modern humans"?
It means that before the point at which the lineage that we call "humans" interbreed with the lineage that we call "neanderthal", these particular pieces of DNA were found only in the neanderthal lineage.