Should a patient with emphysema clean their own apartment?

Depends. If you are sensitive to dust, and dust makes your lungs react, then you shouldn't clean your apartment. But, if you don't have a problem with dust, then staying active with emphysema is always good. That said, you can have someone clean your apartment for you and blame the emphysema. ;) but, i will leave that up to you...
Feasible. Depends on their activity level. If they are able to do it , that would be good. If not, then they need help. Dusting might trigger cough. Avoid chlorinated bleach.