Whether anything in this post will be practicable to most readers remains to be seen, but I thought I'd at least provide some uncommon perspective.
China is, by far, the most polluted country in the world; this is not just in terms of non-lethal substances such as CO2, but also many substances that are extremely hazardous to animal & human life.
Dealing with the water: One should never drink tap water; tap water may either have micro-organisms that cause sickness or heavy metals; many reports come out of China that polluted water is the most devastating form of pollution, causing swelling of kidneys and millions of deaths a year. Tap water also, due to the metallic content, ruins the taste of whatever drink it is in. Luckily, it is the easiest pollutant to avoid, assuming one is careful to always buy bottled water. This is a must, because one can obviously not train at one's peak while suffering from illness or organ failure.
Dealing with the air: In terms of air pollution levels, Beijing is somewhere in the middle; basically scoring average for Chinese air pollution levels--a bad score. While it has been estimated that the water kills 5 times the number as the air, death caused by air pollution is still measured by the unit of million each year. The standard pollutants are chemical, coming from all sorts of power plants and industrial plants that have a mandate to function with total disregard to the environment. Of course, other things occasionally pollute the air, such as styrofoam, the sand of sandstorms, sewer water (due to terrible sewer construction), chemicals & dust used in ubiquitous construction projects, and I even remember one time where a neighboring area decided it'd be more efficient to burn down a crop than tear it up, flooding the entire city with ground-level wind-swept smoke (they also burn trash, tires and so on). Worst still, polluted air is much harder to avoid than polluted water.
It is easy, especially when one is in such an environment for months or years on end, to feel the effects this has on the lungs; having to train intensely and breathe deep merely exacerbates the condition. We are all still learning to deal with this situation, partly we avoid outside training on very polluted days, seek out parks when doing most outside training, try to monitor pollution levels, buy air purifiers for inside use and I have even bought some heavy duty filtration masks which work surprisingly well (I am very happy to have these and I should have gotten them earlier. But they are not perfect, because, especially when heavily exerting the body, they require a slight increase of breathing effort that is necessary to pull the air through the mask to be filtered, making an already challenging training session even more difficult; they are perhaps best for simply assisting one in outside travel on highly polluted days).
Dealing with the food: eating food in China is a triple blow. First, one is not initially acquainted with most of the things available to eat. Second, pollution is a serious concern forcing one to avoid river fish, simply deal with increased metallic levels in the soil that make vegetables less healthy than they should be and also deal with the fact that human fecal matter is used as fertilizer (generally one avoids eating the outside skin of any fruits or vegetables if at all possible). Finally, it is difficult to get a decent amount of nutrients such as complex carbohydrates, which are available everywhere in a place such as the United States. With the increased chance of catching bad bacteria or food already being spoiled, one does not merely risk not getting enough to eat, but also getting violently ill, which has about the most negative short-term impact on training as any other imaginable. Combating this on a budget is the most difficult thing, as a large part of the solution involves buying more expensive imported goods. One also must take pains to find clean restaurants, cook as much as possible, find goods the stomach is familiar with and simply to find enough calories to eat (scarcity of calories is always a problem for the physically active here; when one looks at how anorexicly skinny the majority of the Chinese population is with no exercise, one then starts to realize the challenge of feeding a healthy and strong body that must exercise for hours on end often in less than ideal circumstances).
Dealing with scarcity of space and far too many people:
Beijing is a city of very small apartments and far too many people. As a result, there are certain exercises one cannot train inside and there are very few spacious places outside that are not swarming with overly curious onlookers. If one is not careful, this can have a bad effect on the mind. Training becomes very covert and often limited to very narrow locations. There are two ways, however, that one can find open space: 1. Go to a neglected part of a vast space (these spaces are very hard to find, but they do exist; we have found one such place in Beijing's largest park).
2. If one is willing to have highly unusual sleeping hours and if the air is fresh outside, there are a lot of places that are vacant in the middle of the night that would generally be swarming with people.
I will not get into the terrible summer humidity here, but it is worth noting that with food problems and air problems, extreme humidity and heat compounds an already painful training situation.