Yes, it will run.
Yes, it will run.
I would have liked to purchase a 4670k. Right now I'm running at 1920x1080 with medium settings, and on most maps I maintain around 40fps.
It seems like you're pointing out a discrepancy. The number 8320 likely matters more than 7970 in this context.
I play BF4 at Ultra 1080p without any AA, and I consistently hit around 80fps on most maps. You actually have a faster GPU than me—though mine is customized to run at 1150Mhz. It shouldn't be this big of a difference, even though BF4 is quite demanding on the CPU.
My processor is the 8320, which isn't ideal for games such as Battlefield. With an Intel chip like a 4670 or 4770, performance would be significantly better.