Always safe to have a printed copy of anything but from experience, showing my vaccine certificate on my phone was fine and they didn't give it more than a cursory glance.
What did the plane you caught from Japan say on it's side? At the moment, United don't fly to Thailand, they use code-sharing flights with ANA and Air Japan.
As of currently, there are no flights between CNX and Hong Kong. On booking sites, Hong Kong Express has flights scheduled from November but they haven't made an official announcement yet and that would also depend on any restrictions in Hong Kong. If op wants to transfer in North Asia, Korea would be better and there are regularly scheduled flights to CNX.
United doesn't fly to Thailand - why would you think there is a United Terminal? this isn't the US. You can catch domestic flights from BKK to CNX though with no hassles.