G4560 for 1080p gaming. Upgrade or not?

It depends on what games you're playing. If mostly csgo or similar steam games then more cores won't likely help that much. Having lots of browser tabs open is more a stress on ram, when in the background programs generally default to a lower priority giving the program in focus (a game for example) higher priority. Some games will perform better with additional cores/hyper threading like witcher3, bf1 etc. By better I mean less fps drops and higher min fps which can be noticeable.

Since you have an h series board there's not a lot of reason to go with a k series cpu, the 7600k isn't much faster out of the box than a 7600 (only 100mhz). The 7700k is only 300mhz faster than the 7700. If your monitor is a 60hz then getting fps consistently higher in the 70-80+ fps range won't benefit you much as the panel can still only display 60fps. It depends on what your current fps are in the games you play as to whether a faster cpu will benefit you much. Physical screen size makes no difference, 24", 27" or 40". The system goes by the resolution which is still 1080p and that's mostly impacted by the gpu.

Have you run a program like msi afterburner while gaming to see what your fps look like in game and what percentage of cpu and gpu are in use? If most of your games are at 90% cpu and 100% gpu then a faster cpu may not improve things much. That situation would indicate your gpu is maxed out at the graphics settings and games you're playing. If the cpu is constantly at 90-100% and the gpu is only at 70-80% then a stronger cpu would likely improve things.

Torrent software shouldn't be using a significant amount of resources, mostly drive activity as downloads or uploads are being read/written to the drive. If it's the drive you're gaming from rather than a separate drive it could impact drive related performance. There are other instances where buggy torrent software could cause higher usage, flash ads could increase internet bandwidth and cpu usage or in some cases if infected like anything else extra (virus/malware) could cause higher than usual performance hits. There was a problem a couple years back where utorrent included something called 'epic scale' that was attached to their torrent client purposely. It used the cpu cycles of users to mine bitcoin for their parent bittorrent co.