I have a similar setup to this for my house office. I use an HDMI over ethernet extender (with USB), and put my tower in my computer room in the basement. This way, it is all neat and tidy in the office. I only have a single monitor in there however, a 38" Acer 3840x1600 (@75Hz). I play games just fine, although I never actually tried to see if there was any lag or anything of that sort. The only point of reference I have is my office speakers, which are also wired all the way to the basement and I don't really notice any sync issues with video/audio. Depending on how you perceive visual/audio data however, YMMV.
Also, when I had my house built, I had them use CAT6A so bandwidth isn't an issue. I think CAT5e would still work just fine (depends on run), but anything below might be iffy.
EDIT: forgot to add link
I use this:
https://www.amazon.com/gp/product/B07DFSNT9N . It has all sorts of bad reviews, but as long as you are aware of its limitations, it does work well. I've had it for 4 years now and it's still going.