I've understand that outside of African, most ethnic groups carry some (4% or less) Neanderthal DNA. So en masse, across all living humans, what percentage of the original Neanderthal genome is still alive in human decedents today?
Even though individual humans only have around 2-5% Neanderthal genome, it's not the same 2-5% across people. In total, at least 20%, and perhaps as much as 40% of the Neanderthal genome could be recovered from modern human populations:
We identified Neandertal lineages that persist in the DNA of modern humans, in whole-genome sequences from 379 European and 286 East Asian individuals, recovering more than 15 gigabases of introgressed sequence that spans ~20% of the Neandertal genome
However, there are stretches of the human genome that are almost completely devoid of Neanderthal genes. These include male-fertility associated regions (leading to the suggestion that male sapiens/neandertalis hybrids may be been subfertile or sterile):
Genes that are more highly expressed in testes than in any other tissue are especially reduced in Neanderthal ancestry, and there is an approximately fivefold reduction of Neanderthal ancestry on the X chromosome, which is known from studies of diverse species to be especially dense in male hybrid sterility genes. These results suggest that part of the explanation for genomic regions of reduced Neanderthal ancestry is Neanderthal alleles that caused decreased fertility in males when moved to a modern human genetic background.
And some brain-specific regions (suggesting that those are uniquely human):
Regions depleted of archaic lineages are significantly enriched for genes expressed in specific brain regions ... A large region depleted of archaic sequence spans 11 Mb on chromosome 7 and contains the FOXP2 gene (Fig. 4B), which has been associated with speech and language