Any remaining stutter could well be that BF5 and Squad are just a bit much for your CPU and GPU, but I'm just hazarding a guess at this point.
If you're keeping the same GPU (at least for now) there's no reason to get the 3700X as your GPU is going to be the limiting factor, even at 1080p.
Monitors are often slept on when it comes to rigs, so if it's a decent upgrade over what you currently have then I'd go for the 3600, desk and monitor.
As for if that monitor's any good, it's a TN panel, so very good for fast paced games like FPS, racing due to high refresh rate, low latency and Freesync is a nice bonus.
It won't look quite as fancy as an IPS for slower paced games (RTS/grand strategy etc). Depends on what you're playing really.