Can I just program the chip whilst it's in the Uno, then take the chip out, give the correct pins power, and have it work as it would if it was in the Uno?
Yes, you can. Search on Google for "breadboard arduino". If you do it this way you need a few parts more than just power and the ATmega chip (crystal and some capacitors).
Or do they leave the development board there?
Most do, yes. The Arduino is not just meant as a development platform, in most cases it's the "production" platform as well. Experienced users may just develop on an Arduino and then, as soon as the project is doing what it should, produce a PCB and solder all the stuff on it. But this is the exception, the default is using the Arduino board for the whole project's lifetime.