I'm an American who arrived here by air and got 30 days. Can I extend my stay by 14 days at immigration. Or is the extension shorter? Or does it not exist at all?

No official extension is available but they will normally allow 7 days under consideration for a normal TM.7 and 1,900 baht payment. They might require sight of departure tickets to do so.

As said. 7 days maximum.

Or .... hop on a plane RT to anywhere outside the country and get another 30 when you arrive (visa exempt entry) .. or cross any land border RT and get 15 days.