1. Why make her parents being killed by Klingons part of the story at all? I understand there was a narrative reason for her to be raised by Sarek. But her birth parents could have died in a raid by the Orions, due to an accident, due to illness, etc. They chose to give her a backstory with Klingon family trauma. Why do this and not use it?
They did use it. Burnham's history with the Klingons gave her current relationship with them more gravitas. It was especially poignant in her relationship with Ash/Voq.
It was used when Sarek suggested that her history might be impacting her current feelings about how to deal wiht the Klingons at the shrine. It was also used when Burnham was reminded of by the Klingons on Qu'onos and had to leave the area.
It was her history with Klingons that made her decision to talk Starfleet into abandoning their plan to annihilate Qu'onos all the more poignant.
2. Why is her birth parents being killed by Klingons one of the first things established about her character via flashback - when we see her quizzed about a Klingon attack in the Vulcan learning center and she freezes?
Again, for the purposes of adding gravitas and to establish who Burnham is as a person. As a child, Burnham was affected by the Klingon attack and it showed in this scene. However, that scene was a one off in terms of her reaction to Klingons. We don't ever see her freeze up again.
3. Why does SMG play the following scene - waking up in Sickbay, and then rushing to the bridge - as if she is absolutely terrified?
She wasn't terrified. She rushed to the bridge to tell the captain that she had encountered Klingons in the shrine. I think this is where the fanon associated with Burnham having PTSD started.
The writers never wrote Burnham as having PTSD and we know they had a certain way of conveying that a character had this condition from the way Ash Tyler was written. We saw him display classic symptoms of PTSD first on L'Rell's ship, then art various times following his rescue.
On L'Rell's ship, we see Tyler "freeze", unable to implement his Starfleet training. We see him wracked by paralyzing fear and indecision. He has night and day dreams he cannot control. Granted, part of this is probably brought on by the grafting of Voq onto his body, but a very significant part of his post trauma symptoms are due directly to the horrific physical and mental ordeal he went through.
Had the writers been intending to convey to us that Burnham was suffering from post traumatic stress disorder, they would have shown us Burnham going through some of what we saw Ash/Voq going through.
4. Several people in the pilot episode - notably Georgiou and Sarek - made reference to her "history" with the Klingons.
Simply making reference to Burnham's history with Klingons is no indication that she suffered some ongoing mental illness because of that history.
I dunno. Burnham saying "Admiral, if I may. The ideal outcome for any Klingon interaction is battle. They're relentlessly hostile, sir. It's in their nature." certainly sounds racist to me. Put "blacks' or "Arabs" in there and read it again.
This was a simple statement of fact and certainly is no confirmation of racism. She didn't say "blacks" or "arabs", which would have made her statement a lie. She said Klingons, who do have a violent nature. If you think it is an indication of racism, please explain why.
If that discussion with Ash about hiding in the closet while her family was murdered had no real narrative importance - either to her character or the plot - why include it at all?
I didn't say it had "no narrative importance", it just didn't have the narrative function of informing the audience that she was suffering from PTSD. If you think it does, please explain why.