That’s my basic question. What allows mtDNA to cross the nuclear membrane?
The nuclear membrane breaks down during mitosis.
Thus there is no need to actively cross it while intact, since the nuclear DNA is exposed (and at its most vulnerable for insertion events) during mitosis.
My second question is, once mtDNA is inside of the nucleus, how is it incorporated into the nuclear DNA? What is the chemical process that occurs?
I'm not sure what the prevailing theory is here, but it is likely incorporated during DNA replication, essentially as a mistake in a recombination or DNA repair process. (I'll poke around on the net and see if I come up with anything more specific to mtDNA).
Keep in mind, this isn't something that happens "regularly", so it may not have a standard way of occurring; rather it is the result of occasional genetic accidents.