Freddy Smelts A $50,000 Gold Bar From 29.3 Ounces Of Gold| Gold Rush: Mine Rescue With Freddy & Juan 

I'm pretty sure that's true. Police officers don't have a right to privacy. Multiple courts have ruled that police can be recorded. A building official is just a different code enforcement officer.

It would be an action under 42 USC 1983, 1st amendment Violation of federal rights under color of local law

State by state the laws vary on recording consent laws, some needs both parties to consent, some dont, and yes that includes public officials. And even if it doesnt require consent, its considered decent to not forcibly show peoples faces on tv who dont want to be there. Especially when they have very little to do with the episode. These guys were mainly clerks and such with no power or authority to do anything but relate the decisions of those above them. And clearly they didnt want to play hardball, as soon as he came up with the legal argument about the land use they folded. (there may have been background legal wrangling.)

It varies state by state. Some states are one party consent, some are two party. And even when they legally can record, a tv station that wants to be more than some youtuber prank channel of jerks wouldnt forcibly show the faces of people who dont want to be on there.