Here's everything you might want to know, OP.
https://www.luckygunner.com/labs/5-56-vs-223/
To my knowledge, the only commercial dies you'll find are labeled .223 Rem, and yes, you can shoot .223 ammo through a 5.56 gun. The dogma is you should not try it the other way around, and I'll just leave it at that. 5.56 has the longest leade (and thus the "loosest" chamber), while .223 Rem has a shorter leade ("tighter" chamber). Longer chambers can handle higher pressures and can better tolerate grit and gunk on a battlefield, while tighter chambers build (perhaps unwanted) higher pressures for a given cartridge and are more picky on cleanliness, but generally shoot more precisely, all else being equal. There's actually a spectrum of 5.56/.223 chamberings, for which the article provides a table. .223 Wylde, for example, is a popular hybrid of 5.56 and .223, and is intended for both cartridges. My AR Service (competition) rifle is chambered in .223 Compass Lake Engineering (CLE) Match, which is tighter than .223 Rem to try to squeeze every bit of precision out of a barrel.
There are going to be wall thickness differences between brass manufacturers, hence the same powder loads can generate different pressures with different brass. Keep that in mind if you're using random range brass to reload, as it would be best not to load everything near max charge. Also, military brass (and some commercial brass) is going to be crimped. You'll need to remove that so a fresh primer can be set in. I use Dillon's Super Swage tool, but there are other tools / methods. I also trim all my brass to 1.750" to maintain consistency. The CCI #41 primers you have are my favorite, if for no other reason than they are hard and minimize the potential for slam fires with the floating firing pin in ARs. (Yes, that's a thing and I've actually been present when a few occurred.) They also produce 5-shot, sub-MOA groups for my AR Service rifle. Beyond that, stick to your manual(s) and you should be alright.