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Or combine the two for maximum efficacy! Nitrogen until loss of consciousness, then bolt gun until loss of ...appetite

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A friend was brain dead and lived six days after life support was withdrawn. She was in her 30s and I’m still mad her mother said no to organ donation

So to answer OP's question, we don't know, right?

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I don't mean to be rude, but it really seems like you haven't learned about temperature in a rigorous way (Like you would in a statistical mechanics class). It sounds like you've at least had some sort of exposure to undergrad level quantum mechanics, which is great. But recognize that your knowledge may not apply to this, and consider taking a statistical physics class.

If you did take such a class, you would learn that beta (which is propositional to 1/T) can be defined in terms of the partition function of the system of interest, but the entire concept relies on having multiple particles, (not simply one particle that transitions from state to state).

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It depends how you define respiratory impairment. Nerves supplying muscles of breathing come from differing areas of the spine. Notably as another person mentioned the C3, C4, C4 muscles supply the diaphragm. The diaphragm accounts for roughly 60% of your respiratory effort. Other muscle of respiration include accessory muscles of breathing such as intercostal muscles, neck muscles and upper thoracic wall muscles. A lower cervical spine injury ie. C6 would impair all of those muscle but leave the diaphragm. A person with a spinal injury but intact C6 and below has significant impairment of their breathing. An important consideration is the ability to cough, without being able to effectively cough secretions build up and recurrent lower respiratory tract infections can occur. The main muscle to support coughing are abdominal muscles supplied by thoracic nerves. Other factors include the ability to regulate our sympathetic nervous system. A paraplegic above T6 may have no sensation below T6 such that a painful stimulus will result in crazy amplification of the sympathetic nervous system below T6. This can result in this like acute pulmonary oedema from severe pulmonary hypertension. You’ll be fine cracking your back though, lol.

You don't? Why?

I'm sure you can go on liveleak to find some decapitation video, but from seeing a cow get slaughtered by decapitation myself, the heart definitely keeps pumping blood for a good while

First thing I thought of was back to my old first aid training doing chest compressions to the beat of Bee Gee's "Stayin' Alive".

Doesn't make any sense there are still many processes occuring in a dead body. Like decomposition

Also might not be a thing you do to yourself. I received a "nerve block" for routine shoulder surgery that damaged my phrenic nerve, resulting in a loss of diaphragm function on that side. I now have two good shoulders, but only one good lung.

As long as it was fed oxygen... Yes. I think that's where you're being confused... The brain does play a role in the heart pumping... by keeping the other organs supplying the heart with what it needs.

It's why people that are "brain dead" can be kept "alive" for a while, with a machine that breathes for them. The brains not sending the signals to the lungs to get oxygen in the blood stream to supply the heart with the energy it needs to beat.

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For efficient, painless, cheap and easy just put them in a room and fill it with nitrogen.

You're confidently incorrect. I'm only vaguely familiar with this topic as my dad had heart surgery last year, and I have a tetraplegic friend who has explained some of it to me. But it only took about five minutes of fudging about on Wikipedia to find a detailed explanation of how the heart generates its own impulses: https://en.wikipedia.org/wiki/Sinoatrial_node?wprov=sfla1

The brain and other systems can influence the rate, but they aren't the primary driver. This is why people get arrhythmia - different systems sending contradictory signals.