Impressions after 15 episodes in a weekend

Her captain and friend being killed infront of her?

She was the first officer.

 

I find it interesting that someone as smart as Sarek would give Burnham the information on "The Vulcan Hello". He had to know she was emotionally unstable where Klingons were concerned.

 

This is Sarek. He’s really good at repressing emotions in others by ignoring them.

 

Even insanely smart people have their blind spots. Sarek admits as much in Star Trek III.

 

No we don’t.

 

But they showed that type of irrational emotional response was something that plagued her from childhood (the Vulcan Education Room scene where as soon as the Computer starts asking her questions about Klingon Raids, and the Raid the killed her parents, she reacted in a similar emotional fashion.

When her surrogate mother figure Captain Philippa Georgiou, she wasn't thinking rationally. She saw the murderer commit the act, and in the heat of passion she pulled the trigger and killed him. That was a big part of her character conflict - she wanted to be / and thought she was as emotionless as any Vulcan; but the Vulcans around her could immediately see that wasn't the case and she was still at her core 100% Human, and prone to irrational Human emotional responses.