Yes, .223 and 5.56 are almost exactly the same size.
No, this is an empty ammo case that can hold 400 rounds of your ammunition.
This MTM Ammo Can is outfitted to simply store ammunition. I suggest checking out the TSA requirements for transporting firearms. As far as a nice, hard sided, locking container, I wouldn't recommend this nor would I trust it to withstand the abuse of loading/unloading baggage. I don't know about you, but I've seen how those bags get thrown around.
https://www.tsa.gov/travel/transporting-firearms-and-ammunition
I would recommend a Pelican Vault - Extremely tough, durable, high impact polymer that is crushproof, dustproof, and weather resistant and has secure push button latches and stainless steel reinforced lock loops for your TSA approved locks. The Pelican Vault has 5 layers of high density foam that you can cut out in the shape you need, adding to the safety and security of your items in transit. The Pelican Vault V100 is the same price and much more suited to your need, or check out other sizes if you want to keep any other items in the case as well.
https://www.opticsplanet.com/pelican-vault-v100-small-pistol-case.html?_iv_gun-type=pistol,handgun
https://www.opticsplanet.com/pelican-vault-v200-medium-pistol-case.html?_iv_gun-type=pistol,handgun
https://www.opticsplanet.com/pelican-shooting-accessories.html