Many biologists are now classifying modern humans and Neanderthals as related subspecies. The taxonomical assignations are then Homo sapiens sapiens and Homo sapiens neanderthalensis. There's also evidence of cross breeding with Denisovans, but they haven't been properly classified, yet, and may not be a separate species or subspecies.

The percentage will have been higher among the first tribes to breed with Neanderthals, but probably have fallen to the current level fairly quickly. Generally, "weird" qualities will disappear, unless they're useful (not to mention if they're disadvantageous in their new context), which will reduce genetic variation. This has removed much Neanderthal DNA. Only a few alleles are shared in modern populations.