Today’s expressions

Sounds ok! But people normally  dont use this expression when they speak - just a point to remember. 

(2) at the expense of
- My mom really worked hard to raise me up at the expense of her life.
- I don’t have the courage to have a baby at the expense of my life, because I believe that if I’m not ready for a baby, I’ll drive their life to the wrong way, which I don’t want to happen.

This one is not correct. 'at the expense of my life' means that you sacrificed your life (aka you're dead now). 

(3) entitled
- Since our society is customer-driven, some of people act as if they’re entitled.
- A lot of high ranking officers tend to have a sense of entitlement just because they think they should be treated better than ordinary people due to the power they have.

Sounds good

(4) edge
- what should I do to have a distinct edge over others in a job market ?
- Try to write a journal as often as you can. It’ll give you an edge over people learning English.

distant edge (X), edge (O). just say "what should I do to have an edge over others in a job market ?". it sounds better like how it's used in the second sentence.

(5) shudder
- Just imagining the terrible things they had to go through makes me shudder
- I would definitely shudder if I walk through dark alley and see a stranger’s shadow or hear someone walking behind me.

Sounds fine

(6) ruthless
- I don’t wanna be a type of person who’s ruthless to get ahead.
- Being ruthless to be successful can be justified ?

Sounds fine

*Anyone who helps me out,Thank you in advance :-)