Regarding the quality and accuracy of the Neanderthal detection percentage I believe they have it pretty much sown up. They have very good samples of DNA and compare that with samples taken from volunteers. The denisovan percentage is still in an experimental phase as they only very recently discovered, identified the separate species DNA.
Dont forget that the male DNA, the Y chromosome is passed unchanged from father to son, unchanged, generation to generation. Evolution being what it is, clever as it is, we only keep what we need, what is useful to survival. Of course we still get some of the not so useful DNA passed too, disease, weaknesses etc. I too think that we owe lots to our neanderthal cousins, we wouldn't be here otherwise.