Yes, but that doesn't answer the mechanism by which it crosses the nuclear pore, or integrates with the nuclear chromosomes.
Again, it still happens today; numts are common. Linear DNA strands are easily thin enough to fit through a nuclear pore, and "homologous recombination" (a misnomer) is almost universal.
In vertebrates, mitochondria have a modified genetic code. Therefore, mt genes that end up in the nucleus are automatically pseudogenes. But that's not the case in many other eukaryotes.
The nuclear envelope disappears during mitosis -
Not in all eukaryotes by far. I don't know what the plesiomorphy is.
All of the above?
Probably all of the above at different points.