This is simply brilliant. It's quite short and simple, but it's so effective. And Galbatorix's story is really sad.

So, this is my attempt to offer a possible point of view Galbatorix may have had on canon events.
Although it's not my personal view, it certainly is something that could have happened and I accept it as canon for another world, if you know what I mean.

I hope I've presented these topics without sensationalizing them or disrespecting folks who suffer from mental illness.
For me, I can certainly say you have done well. Just by presenting it from his point of view, you do justice to his experiences.

The first time he heard a voice he definitely knew had never been spoken, he was holding Jarnunvösk’s lifeless head.
An interesting way to refer to the voices he hears. And I feel sorry for Jarnunvösk, even though I barely know her.

Oromis-elda may claim that there was no afterworld, but he didn’t know.
Let Oromis claim what he may. He should stop talking about things he doesn't know anything about. I mean, I still don't know whether or not Alagaësia would have an afterworld. And I also see you implied Oromis was Galbatorix's teacher, which is quite heavily implied in canon.

He carefully nicked one of his fingertips - enough that it would bleed, not so much it’d be a danger
Interesting. This really is the good kind of worldbuilding.

He reached out with his hand and drew a rune on Jarnunvösk’s cheek
This is really touching.

“She was more worthy than you.”
This line here is so sad. I know he will never think otherwise, although I would want to tell him it's not that way, and I know I would think the same.

He forced his numb limbs to move to his companions and without hesitation took their debts upon himself.
Because he cares about others.

The 92nd time he heard a voice he definitely knew had never been spoken, he’d been walking south for some time.
It's so sad he's counting the number of times he hears the voices, and that he carries on for so long.

Though he was among trees now, he told himself he needed an axe first to pay the debt.
...And he's postponing. And I still know he's doomed.

“Don’t the dragons choose?” The voices whispered
Sure they do.

Shruikan would never grasp that not all of the threats his Rider saw or feared were real, and others would read this as cruelty and a disposition towards violence.
Not exactly as I picture him, but he still is mostly the same. He finds he must protect his Rider, whatever the cost may be.

When they spoke of Jarnunvösk, the two spoke in the language of dragons - memory and impression.
Just as the language of dragons, or indeed, most other animals, should be.

I'll comment further tomorrow.