She cannot be killed and is therefore invincible. It's pretty simple, but Mokou has effectively attained eternal life having tasted the elixir of immortality. Most folk think of 'immortal' as just a mortal that's really really hard to kill, but I like how it's true meaning is revived when played in IN. Mokou quite literally is missing death, which is why she scared the daylights out of Yuyuko. I don't think the nature of it is regenerative like cell division, rather her existence endures because there is no possibility for an alternative. I guess with her there is no defining line between body and soul, which represents life and death. In IN, she even explicitly declared that she's free from those boundaries.
I think it's already been discussed, but I think Keine's abilities regarding history are based on substituting timelines with illusionary realities. If you think about it, everyone, including Keine, have memories of the events Keine erased or rewritten. Yes, the village was gone when she hides it, but Yukari easily saw through it and found the real one 'underneath it all', seeing as mingles with borders between reality and fantasy all the time. And Keine has to have memory of the things she undoes or rewrites. The grandfather paradox demands the real thing exists somewhere, wherever it could be.