Buy the best barrel you can afford.
The AR 15 is a rifle, not a caliber. the .223 is a caliber. Usually the AR-15 is made in caliber .223.
No.
Ar-15 for one. You can use .223 ammo in 5.56 barrells as well.
600-1200
No. The two rounds are not interchangeable.
ar-15
100-1000 USD or so
.223
Proofhouse.com has sn data.
I believe, for the most part, cal 223 or 5.56
50-1000 usd
Depends on your configuration. The most common, by far, is .223 Remington/5.56x45mm NATO.