Well, the way I understand it, the matrix links itself into essentially every aspect of your brain, so that your brain isn't really sending signals to your body anymore, only the matrix, and vice versa. So, your brain would not recieve signals that are saying that you are hurting your own body, only the signals from the matrix telling it that you are in fact hurt. And if you don't believe how powerful is, it really can kill you. Although maybe the blood in real life I can't really explain (but I'll try in paragraph 3) it is quite easy to manipulate the body using the mind. A simple example is headaches. With some concentration, you can convince yourself the headache is in fact an illusion, and that you are fine, and it really will take effect, I do this often.
So applying that to the matrix, one could argue that somebody could simply allow themselves to be shot in the matrix and just think, "It's not real, I am fine" but I would theorize that because the matrix has a direct link to your most vital nerves and brain, it would require an intense amount of concentration no human would have to override such direct signals from the matrix. As opposed to general effects, such as gravity. That's why laws can be 'bent' but not broken. Although Neo has boundaries lifted off of him, he can override more of these signals and therefore do extroardinary things.
This also leads to a theory on the bleeding, although I can't back it up very well at all, it occured to me that if your brain does send subliminial, instinctive signals that sneek past the matrix, that may be the reason right there. Like, when you are cut, and a scab covers it up. But what if a scab tried to build without a cut actually there? Maybe it would cause the, or a similar, problem. But like I said, I can't back that one up.
So, to use your suffocation example, it would not be the brain making you suffocate. But the brain would recieve signals from the matrix simply telling it that you aren't breathing. So, even if you really are breathing fine, all it would take would be for the mind to belief you aren't and it would shut down, killing you.
Maybe these don't add up to one evidential conclusion, but these are just my thoughts on the subject. But it leads to other questions, like is there anything Neo couldn't do? If not, why would he not do more? And yes, similar questions towards machines, except that where humans are bound by the matrix signals, machines are also bound by the way they were coded into the matrix. A lot to think about.
My final thought is how differences affect redpills compared to bluepills. Because bluepills have jacks not only inn their heads, but also other essential parts of the body, maybe this prevents things such as bleeding inside the pod? Whereas redpills do not have those extra jacks, so maybe their body feels it directly instead. But, as we see in World Record, humans do have the potential to become concentrate so hard, that even the matrix can't hold back the signals, as we see the runner actually begin to run inside of his pod, severing the connections.
Very deep stuff.
