it's generally not thought that H. erectus evolved into Neandertals/Denisovans outside of Africa. Erectus was already outside of Africa long before the split separating the N/D lineage from ancestral sapiens happened within Africa.

Neandertals/Denisovans likely descended from H. heidelbergensis. A similar looking population to heidelbergensis is found in Africa, where it is sometimes called H. rhodesiensis. This population leaving Africa maybe ~700,000 years ago likely gave rise to Neandertals and Denisovans in Eurasia, while that staying behind (rhodesiensis) could have given rise to sapiens.

For comparison, Erectus is found outside of Africa (in Java) by at least 1.4 million years ago.

Interestingly, the genome of a Denisovan in Siberia showed small traces of an even older genome. This may have been from interbreeding with erectus.

1. all sperm cells do not carry the same DNA, they each carry a different mixture of the two sets of genes the father got from his two parents (everyone has two copies of their genes, one from their mother, one from their father).
2. all egg cells do not carry the same DNA. As above, they carry a random mix of the genes from the mother's two parents.
3. the chance of any two sperm or eggs having the exact mix are so low that this never really happens. this is why children from the same parents are not identical.

should definitely get a nice match with a mother, but i wouldn't put anyone on death row based on that alone :)

When you use DNA to match to a suspect, you usually use short tandem repeat (STR) length patterns in nuclear DNA, which change more frequently than nuclear DNA by several orders of magnitude.

On average a mitochondrial sequence will have a single mutation every several hundred years. So exact matches are common if it's within ~20 generations or so.

All those "rules" you cite are just conjecture. No one knows the fertility of Neandertal and sapiens offspring.

An adequate reason why there is no Neanderthal mitochondria in modern humans could be there was only a small amount to begin with and it was lost over the many generations since then. A small amount would have high odds against it to last very long.

This is the clearest overall response. The system described will not be able to boil the water using only the energy from boiling the water. Let alone powering the ship as well.