No one has any DNA that can be traced to a specific Neanderthal, no.
Of course people have DNA in common with their ancestors as a group, hell we share a lot in common with fish. But it's because these genes are common that they enter and re-enter your family tree at multiple points.
And your mitochondrial DNA definitely doesn't descend from Abraham, a male incapable of passing on mitochondrial DNA, so there's a reason I gloss over female line descent.
Your pseudoscience above would make it seem nobody has any DNA from any ancestors at all, as there were not your 10^Whatever creatures ever in existence. Once you figure out the bug in your logic there, you will perhaps see the failure of the rest of your argument. Hint: inbreeding.
It's actually called pedigree collapse, but it doesn't really change anything - the fact that the same ancestor may appear in multiple places in your family tree makes it harder to trace a gene back along any given line, not easier. You undoubtedly share an enormous amount of genetic similarity with Abraham, it's how you got it that is forever scrambled.
Of course people have DNA in common with their ancestors as a group, hell we share a lot in common with fish. But it's because these genes are common that they enter and re-enter your family tree at multiple points.
And your mitochondrial DNA definitely doesn't descend from Abraham, a male incapable of passing on mitochondrial DNA, so there's a reason I gloss over female line descent.