I don't doubt that most courtyards would have these problems to a greater or lesser extent, quite separate from city air issues. Most of that stuff is allergenic, and many molds are actually quite toxic. Not something to take lightly, especially if you have just started experiencing more significant breathing/coughing problems since you moved there.

Frankly, I think there is no good way to remediate this and particularly not in courtyard housing. You don't get rid of mold by fumigation anyway--it has to be removed (along with the materials propagating the mold), and then completely decontaminated. Depending on how extensive it is behind the scenes (where you can't see it), it may take a major rebuild of the place to eliminate, or it will just come back. I strongly suggest you consider moving elsewhere in Beijing, asap to a newer place with as few soft surfaces (i.e. carpeting) as possible. In summary, GET OUT!