Does the "nature" of a person move with her to the next life? Imagine a person that is very calm, generous, doesn't like to harm others etc. If this person gets a rebirth in the human realm will he/she develop the same characteristcs? 

The point is that Kamma and the last mental state can go on, but according to Anatta I'm not sure about the rest.
kamma doesn't actually go on. kamma means action, and it is finished after it is performed. Nothing actually goes on from one moment to the next; the effects of one moment are felt in the next, and have potential repercussions far into the future.
It is these repercussions that do indeed create a sense of continuity from life to life; a person who ended their life with a certain nature will be reborn with a similar nature, simply due to the causal nature of reality.

&quot;Bhikkhus, whatever a bhikkhu frequently thinks and ponders upon, that will become the inclination of his mind.&quot;
MN 19 (Bodhi, trans)