While the external case dimensions are very similar, the .223 Remington and 5.56x45mm differ in both maximum pressure and chamber shape. The maximum and mean pressures for some varieties of the 5.56 mm (different cartridge designations have different standards) exceed the SAAMI maximums for the .223 Remington, and the methods for measuring pressures differ between NATO and SAAMI. The 5.56 mm chamber specification has also changed over time since its adoption, as the current military loading (NATO SS-109 or US M855) uses longer, heavier bullets than the original loading did. This has resulted in a lengthening of the throat in the 5.56 mm chamber. Thus, while .223 Remington ammunition can be safely fired in a 5.56 mm chambered gun, firing 5.56 mm ammunition in a .223 Remington chamber may produce pressures in excess of even the 5.56 mm specifications due to the shorter throat.
So you can shoot both .223 and 5.56 in a rifle chambered for 5.56. You SHOULD only shoot .223 in a rifle chambered for .223.
As for breaking in a barrel; shoot once, clean, repeat 5ish times, shoot two, clean, repeat5ish times, etc up to about ten rounds. I'm not an expert, but this sounds right, guys?