2. The PA (physician assistant) made a mistake. Hemoptosis should be the symtom s/he should have written down. You can either point that out to the PA by phone or on your next visit, or contact his or her supervising M.D. to clearify.
3. Looks like the PA was trying to treat allergy/dry nasal mucosa first to see whether the blood in sputum will go away. And if bloody sputum persists, they will further investigate. That is all the general practitioner, including family physicians and most internal medicine physicians would do.
Hemoptosis (coughing of blood) or even blood in sputum should be taken seriously (not like most of the posts in this forume), especially for Chinese -- need to rule out nasopharyngeal tumor or lung lesions.
Hope that helps, and good luck!