Yup, if you have serious pneumonia, mucus comes out almost every orifice and can stop up the airways. If he has a part of his beak missing, has to open his mouth to breathe better, has stuff blocking up his nostrils, and has just come away from near death due to pet store problems, malnutrition, he is fighting for his life. The part where you say you can't hear any noises could mean that his airways are way too stopped up to make much sound anymore. Hurry. Also, his noises may be so quiet that you can't hear them without a stethoscope.
Since I can't see him; since I am not a vet; since he has possible necrosis with his beak which could be due to secondary infection from all he has been through, including a lung infection, ahhhhhhhh, I'd take him to a vet as an emergency. I don't think that his body is going to get well by itself. Antibiotics are developed to attack bacteria. Bacteria have cell parts that are so foreign to our own body's cell parts, that the antibiotic can target bacterial parts and not hurt our own. The antibiotic could be a life saver for your turtle. Some bacteria kill us animals before our immune systems have a chance to put up a big enough fight. And, your turtle came to you malnurished, meaning he didn't have enough of the amino acids/proteins to put up a good fight, even if the turtle next door could have. Antibiotics are a way to give your turtle missiles when the attacker is using machine guns. He may be, probably is in pain. Please hurry.
It is what I would do. If it is cold outside, protect him by putting him in a cloth, put him inside your coat, against your body warmth and hurry.
Being safe ends up feeling good. Being too late ends up hurting every time you remember it. Please be safe with this little guy.

