You can't quote me on this, but I don't think there are any vaccines required for visiting China.
What part of China are they going to?
The pollution isn't great; whenever I go anywhere other than Western Europe I find myself blowing my nose with some black stuff in there. Ick. Happened in China, Serbia, and Tanzania. I think it's mostly the diesel. However, I don't believe that a few weeks exposure has shortened my life any.
Avian flu is a concern worth considering, but currently it is not spreading from human to human. Your child will not be butchering chickens, so the risk is VERY low IMHO.
In major cities (Beijing and Shanghai for example) the government is very careful with tourists. In Tiananment Square I was taking a picture and turned around to see a guard yanking away and yelling at a man who was a little too close to my backpack. I kept my guard up more after that but also felt good about the protection there.
I don't think a 10 year old would be capable of keeping their guard up like that, and would need an adult to be on their toes at all times.
Poorer people will swarm foreigners to sell trinkets and such. It can be overwhelming.
Middle class people are very welcoming of Americans and curious.
If you're white or black (that is, not Chinese descent) you'll get stared at everywhere. It can be disconcerting.
I would not let my 10 year old go with her father to meet his online girlfriend ANYWHERE.
I would let my 13+ child go with a school trip without a parent (after asking questions and getting info about itinerary, precautions, rules, etc.).
I would let my child any age go with a trusted AND capable parent (that is, one sophisticated and aware enough to handle the pressures of foreign travel) ... but not to meet an online girlfriend.