NYTrever- Regarding the emissions Inspection failures; ("Then, after another 50 miles or so, the report, sometimes a day or two later, at times the same day, shows the car passing the inspection").
Cars often require driving 50-100 miles after maintenance—especially after battery disconnection or code clearing—
to complete a specific "drive cycle" that resets the On-Board Diagnostic (OBDII) monitors to a "Ready" state. This process ensures the computer has monitored engine performance and emissions systems, verifying that no faults exist.
-CFK