Single Eve: still holds; given the small population size of humans in the past, at some point there was a female ancestral to us all, as other lineages would have stochastically died out.
Human dispersal and Neanderthal-modern human crosses: both have been examined via autosomal DNA. Reconstructions of dispersal may be skewed in the following manner: since we use X- Y- and mtDNA to look as sex ratios of admixture events (such as in modern Caribbean populations, where there is more European male influence and African/Native American female contribution), our reconstruction may be skewed by these selection due to incompatibilities. We expect there to have been some incompatibilities between modern humans and Neanderthals/Denisovans—no doubt some were mitochondrial. But autosomal DNA bears the evidence of our admixture. Likely some lineages (including those with prehistoric-human mt- and Y-DNA) were lost, either due to selection, stochasticity, or likely, both.