The question of whether the Neanderthals were a separate species is really a moot one. We define species by the (arbitrary) standard of whether members from two populations can produce viable offspring, but that's not something you can apply to extinct populations. The only answerable question is whether every individual can be unambiguously assigned to one population or the other, based upon morphology. But my understanding is that that's been the case for a long time.
This is mostly true but not totally. The Lagar Velho child was originally thought to be a hybrid. Some now argue that it only shows that very young Neanderthals were less robust-looking and different from us than the adults. There are also some specimens from 90-100kya, the Skuhl site in Israel, which are overall classified as early modern. Nevertheless, one or two have some "Neanderthalish" features which may result from then-current crossbreeding with Neanderthals or may just represent "atavistic" variations within that population.
DNA studies with nuclear DNA would be best, but nuclear DNA is much harder to recover than mtDNA.