Are there aspects of the mind that owe nothing to the physical functioning of the brain?
Are there aspects of the mind that do not correspond to any physical area of the brain?
What exactly does the soul do that confers consciousness on us, and is this influence transmitted to the brain?
Does an immaterial soul interact with a brain and body of matter? Does the soul control the body at all?
If so, how exactly does it do this? What force or causal influence does it exert, and by what mechanism? By what mechanism can an immaterial soul possibly alter the state of the body or affect the brain?
Neuroplasticity. Intentional behaviour changes the matter in the brain.
Why would God create an immutable/unchangeable soul-nature and then make it subject to the changeable and imperfect nature of a fallible material body, and judge it for the actions committed by that body?
Because He wanted this way. The judgement is related to the intentions of the immortal soul and not the body.
Why do we even need such bodies, if at best they can only allow the true nature of the soul to shine through unaltered, and at worst obscure it?
The human body came first by evolution. The human nature is designed to be a double nature.
What is God’s rationale for imprisoning souls in bodies and holding the souls responsible for the uncontrollable irrationalities of those bodies?
The original plan was the soul having perfect control over the body. And humans are not imprisoned ghosts and are not only physical bodies. Human nature is the union of spirit and matter. The imperfect union between soul and body is temporary and the soul without the body is a temporary state too.
Which animals are these?
Globicephala melas: "For the first time, we show that a species of dolphin has more neocortical neurons than any mammal studied to date including humans
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4244864/
The evidence shows that our identity, personality and behaviour are completely determined by the physical configuration of the brain, and that a change to this configuration can alter or eliminate any of them.
The soul needs the brain to be functional in order to interact in the material world. But brain damage doesn't have any effects on soul. It's just like a broken radio can't do a good reception from the broadcast. But it doesn't mean the malfunction of a radio set affects the broadcasting source.