I have a shady yard and even shadier home on the inside. I know a tiiny bit about what will grow in shade and poor light. I also have DD's with severe allergies to the world and deal with high humidity indoors as well as outside .
Plants first: My Peace Lily grows well at a north facing window and is one that cleans the air. It was the only one out of many that I chose to keep indoors after winter. It is now in a huuuuuge pot. But I know you can buy them in small sizes for your son's shelf. ZZplant needs more light than dense shade but on a north facing window sill it would do nicely. Cast Iron plant will be a bit larger but it cant stand almost no light. There are variegated varieties but they need a bit more light. They are drought tolerant, too. Japanese schefflera can take some shade. But will eventually grow into a tree if not a Bonsai.
There is something out there called Damp Rid@Walmart etc... They are crystals that absorb moisture. My DD's are allergic to mold. We use these in obvious humid places, bathrooms laundry room etc. and also in their bedrooms.There are inexpensive electronic mold-killers online as well.
We have air cleaners in all our rooms and two or more in large areas where our pets are. Stay clear of Ozone generating air cleaners that zap the dust. They leave behind soot that your son may also be allergic to which defeats the purpose. When a HEPA air filters says , "Cleans a room 8 x 10'" or whatever size, that means the air will be recirculated and cleaned many times while on the HI setting. If you don't want to hear the hum of the machine on HI, you will have to get a filter that is larger than the room measures to keep it on low or med. The black charcoal pre-filter can be vacuumed monthly despite that the instructions say to replace it. They just want to keep selling them to you. But a good cleaning will keep them clean for many uses. Also, keep the hampers that contain dirty, pollen-filled clothes out of the bedroom. Cover both top and bottom mattresses to keep dust mites from contaminating the room. House books,stuffed animals and other dusty toys in covered plastic bins. Keeps toys organized and keeps the dust from settling on them.
I have many more hints about your home but most of which you can find online. We never got rid of our pets or replaced our brand new carpet like the doctor asked. Our DD's are thriving in the real world without being over protected in a sterile environment.
