Again, you presented the DNA argument as and end all pillar of truth, but it cannot even stand rudimentary questions.
You didn't anwer one of them.
What Neanderthal DNA looks like? How it was collected? How did you arrived at the notion that this DNA sample is what you say it is? Where have you obtained this DNA?
The key, dear SkyPilot, is to continue reading past that right-up-front sentence to the point where the questions are answered.
The heck, can't you read? That link clearly states that the evidence shows there has been no mixing of Neanderthal mitochondrial DNA into the human population. Yes, they say they are difficulties, but they clearly feel these have been surmounted.
What does Neanderthal DNA look like: Like human DNA or chimp DNA, but with significant differences.
How was it collected: Extracted from Neanderthal bones and amplified by PCR (hint--perhaps if you don't know what PCR is you are not qualified to make a judgement on the validity of this study).
How do we know the sample obtained is Neanderthal DNA: Because it came from Neanderthal bones and shares many similarities with human and chimp DNA.
Where was the DNA obtained: A variety of European sites.
The mitochondrial DNA argument can indeed stand rudimentary (perhaps "elementary" is a better descriptor) questions. Just because you don't understand it does not mean it is wrong.
If you really want all of your methodological questions answered in depth, I suggest you look up the original article and any supporting information provided for it.