This may be kind of obvious but I found it really interesting.
Other than that, I don't think Rem could die.
I don't know what's gonna happen but...
She says it clear... what she is about do will not mean anything to her. It will be just another dead human.
I believe in order to kill a shinigami... the shinigami has to save their life out of love. Rem might in fact have love... but it remains to be seen if she would actually let it motivate her actions (and not just her words).
What she does at this point in the story is neutral as far as she is concerned and she's just doing what a normal death god (presumably) does. Scribble in their notebook.
aka KILL.