The TV series takes place in the same world as Witch Hunter Robin."Neuros" and "Witches" are different names for the same thing: people with Superpowerful Genetics, many of whom are abusing these powers and gaining the attention of extra-legal agencies who hunt them.
Jane got her powers from a Neuro.There was an episode about a Neuro who could give normal humans super-powers. The one that tagged Jane was just a little better at it.
Jane is a Neuro, and one aspect of her powers is to mask the gene from detection.
Jane is an alternate dimension version of WolverineBadass, fast healing, does things her own way.
Jane is experiencing Quantum ImmortalityQuantum Immortality is a theory linked to multiple universe theory. Basically no observer can observe it's own death. Since there are an infinite set of possibilities expressed an observer will always and only be aware of those possibilities in which it survives. Even in the event of the chance of survival being infinitely small it is is still expressed someplace along the continuum of multiple realities.