It does not seem we understand the mitochondrial inheritance mechanism during conception.
If all paternal mitochondria were transferred every time (unlikely‡), then a child's mitochondria would be ~99.99% inherited through the mother.
‡ Unlikely because sperm mitochondria are located toward the flagellum in order to power locomotion, and the capsule at the head is normally the only part which fully penetrates the cell membrane.
Ref: https://www.quora.com/How-many-mitochondria-are-there-in-hum...
"Although biparental inheritance of mtDNA and heteroplasmy coincided with disease symptoms in some of the individuals studied by Luo et al., the authors’ data do not demonstrate a causal link with disease. In fact, we cannot be certain that the study participants have mitochondrial disease, because no specific examinations to confirm this diagnosis are reported. Further study is needed to identify more cases of potential paternal mtDNA inheritance, and to determine the functional consequences of such heteroplasmy."
So no, it is not definitively linked with mitochondrial disease. If anything, the article suggests that the inheritance may be due to an abnormality in the host DNA:
"Previous work has shown that mitophagy, the process by which cells ‘eat’ their own mitochondria, has a role in the selective elimination of paternal mitochondria. Given our rapidly expanding knowledge of mammalian mitophagy in vivo16, these rare instances of paternal mtDNA transmission might be attributed to defective mitochondrial turnover. The inheritance pattern of paternal mtDNA in Luo and colleagues’ study suggests that a yet unidentified gene on one of the autosomes (non-sex chromosomes) is involved in eliminating paternal mitochondria."
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.