I think you have a few errors in your explanations.

There is not enough evidence about the immune system of the neanderthals. When the Out-of-Africa migration(s) happened, there is no evidence that the neanderthals had encountered significant diseases and epidemics in the past -- up to that point. This does not rule out the possibility that they neanderthals were spreading some unique parasites amongst themselves. Disease is all about congestion [1].

What the neanderthals had that was most beneficial to humans was geographic adaptation. And the are genes for that can jump-start (by many generations) the evolution of features like longer and more straight hair.

Also, during Natural Selection genes that become useless will stay in the gene pool. Natural Selection selects for advantageous genes [2], that does not mean that it selects against neutral (useless) genes. DNA has no size cap and the gene pool of organisms continues to hold onto the genes long after they offered a survival edge [3].

[1] It is also congestion that increases the likelyhood that a disease jumps species.

[2] And by compliment, against disadvantageous genes

[3] This is a virtue of DNA, it likes to keep a memory bank of all solutions that have been useful in the past. It is a hoarder.

Just to elaborate, because eukaryotic genomes are so huge the marginal cost of copying excess genes is negligible so the selective pressure against them is as well. But prokaryotes have very small genomes and reproducing useless genes would be relatively disadvantageous for them so there is noticeable selective pressure to eliminate useless genes for them. So you will see the frequency of unused genes in bacteria tend to decrease over time.

> Also, during Natural Selection genes that become useless will stay in the gene pool.

That is absolutely true, but useless genes have a much higher probability of getting corrupt by accumulating errors, because the lack of useless genes is not selected against.