.223 is an English measurement,,,5.56 is a metric measurement. In the USA, since we are not a metric country, the round became the .223 Remington. However, as a founding member of NATO and trying to work with metric countries, the round known as the .223 Remington is called the 5.56 x 45 or the 5.56 NATO. This is the same as the .308 Winchester being known as the 7.62 NATO.
As far as breaking in a barrel,,,you can either do the benchrest route,,fire 1, clean...fire 2,,,clean, etc or do the Derrick Martin route,, which is to go out a fire a 100 rounds and clean it. The bottomline is do what makes you feel comfortable. I have two service rifles, one I did the Benchresh break in,,,the other I did the Derrick Martin route,,,,both will drill a .5 MOA hole.
In any case, going haywire and shooting a ton of rounds in rapid order, overheating the barrel, can burn the throat out.
A good reasonable cost ammo is Black Hills remanufactured ammo. If you run out and need to buy ammo internationaly, just make sure what you buy is 5.56 NATO made for military customers.