You will be fine with that monitor with bf3/4 just turn on adaptive v-sync in the nvidia panel and you should be maxing out the frames with almost any game. That monitor is really made for both PC/console but is a monitor at its heart, so it will basically run the same as any gaming monitor. Hope that helps
not all 600w PSU's are created equal though. Make sure to look at total watts on the 12v rail(they should be very close to the max 600w) also make sure to look at the amps. Most 770's require a min of 42 amps on the 12v rail. Also really when they say all of this its based off of total system usage.
Pretty sure my mom is using a pre built ASUS i7-2600 with 8GB RAM with a 22 inch 1080p monitor, beyond that I have no idea. They opted to have a pre built for the warranty ease. Otherwise I would have built one for them.