The Atmega328 has an operating voltage of 1.8V to 5.5V. If I was operating it at 3V (by supplying 3V to VCC) and set a pin to high, would that pin be at 3V?
The high output voltage of a pin (VOH) depends on the operating voltage (VCC), but it also depends on the current you'll draw (source current / IOH) from that pin. Take a look at the ATmega328 datasheet, page 411:
You can see from the graph above that the more current you draw (source) from a pin, the lower its output voltage will be. For example, if you connect a LED to an I/O pin and that LED will draw 15 mA then the output voltage of that pin will only be aprox. 2.4 V.
OTOH, if the pin had no load connected to it or a very high impedance load, its output voltage would be near the VCC voltage (3 V).