A couple of additional questions, Nrdsb4, if you are still willing to provide expert information:
Reading at the Wikipedia level indicates that the release of adrenaline is controlled by the autonomic nervous system, so this could be a spinal-cord level of activity. Is that right?
Release of cortisol seems to be controlled by the anterior lobe of the pituitary in response to ACTH, which is produced by the hypothalamus. It does not appear to me offhand that the release of cortisol would occur for someone with no brain activity. Is that also right? Or perhaps there is another path, under the exclusive control of the spinal-cord?
I can see how ventilator setting, if they were changing, could affect the heart rate; but if they were unchanging, I would not expect the heart rate to change. I’d guess that the same is true for body temperature. An increase in body temperature might cause an increase in heart rate without nervous system involvement at all (but I am not sure about that). Are these guesses correct?