Most multi-engine airplanes can even take-off and climb with only a single engine functioning. You would never do it intentionally, but sometimes engines fail shortly after takeoff when you are 50 feet above the runway.
This is likely true. He was the only person onboard the plane, so he would not have been able to get up and look. Unlikely he could see the extent of damage from his seat. And as noted, on final approach, he doesn't have time for much to change plans. Even if he knew the extent of the damage, the choices are roughly the same - land as planned, or go around.
That said, he did exactly what he should. Aviate, navigate, communicate. He controlled the plane, made a decision, and communicated that to ATC. Well done.
It is not so black and white. One might (and normal person should) feel a stress reaction with rising levels adrenaline and suchlike in his bloodstream, and it is normal, it might even be beneficial, if one was trained to benefit from this mind state. I become more resolute from this, more concentrated than usual, sometimes even more concentrated than it is good for me because I might miss some important detail.
> If a tyre blows out, most of us wouldn't scream and close our eyes, we would try and steer to safety.
I never was in this situation with a blown tyre, but I was in some other situations when there was a danger to my life (or I believed that it was) I become extremely concentrated, almost silent (except for occasional curses), but when I needed to communicate, I did it loudly and with a good deal of swearing. I think that communication for me in these situations is like a distracting hindrance, so if I need to do it, I'll do it in a way that will make it unnecessary to repeat it.
The communication of the pilot have not a hint of his emotions. My mind was blown not because of his perfect actions (I expect perfect actions from a professional), not even because the lack of swearing. It is the tone of his voice, it is completely composed, almost relaxed. I can imagine myself in this situation restraining from shouting and from swearing, but my emotional tension would leak through my voice, I wouldn't even try to hide it.
When you get your medical clearance, one of the things they look for is signs of psychological issues or instability. Not saying that's you, and I bet you'd do better than you think. Pilot training also IMHO makes you better at that stuff.