Everything from the Health Interview is self-report. There was no confirmation, diagnostic testing, nor medical chart review done to compare with what the participant marked on the questionnaire.
#1 - I'm guessing "COPD" is a term less people had discussed with their doctors. SHHS1 was almost 25 years ago now so I'm not sure how these pulmonary diseases were conceptualized and presented to patients back then.
#2 - I think you'll want "asthma15"/"asth1215" for SHHS1 and "hi201d"/"hi201e" for SHHS2. In SHHS1 we only know if they ever received a diagnosis and whether they had an asthma attack in the last 12 months (asth1215), which you might use as a proxy for "having asthma at the time of the PSG".
#3 - The Health Interview questionnaires should have been done around the time of the actual PSGs in the SHHS visits. So, I expect it's on the order of days. (Comparing "stdatep" and "formdate_hi" could be informative; at a glance they look to match as I expect.)
#4 - Unfortunately all we have is self-report. I expect the parent cohorts have more definitive information about testing, diagnoses, and ongoing presence/treatment.