No, originally she didn't know he was her son, which is why she was confused as to why she hesitated when she was about to kill him during Arcade's training sequence.
In her first appearance in the X-Men, she clearly DOES know who Kurt is, and is the one to tell him to ask Margali about what really happened the night she took him in.
As for her killing other people; does killing someone automatically make you irredeemable or something?
When they are no threat to you and you don't seem to care, yes, it pretty much does.
How many decades of repeated, cold blooded, unrepentant murder would it take to convince YOU that someone had no intention of ever reforming, despite repeated offers?