I forgot about PSU, it's 550W but I will change it when I switch GPUs most likely.

Power rating doesn't necessarily mean much. I'd trust a quality 400-450W much more than a cheap 550W.

Also would going up from i3 to i5 on the same socket improve much?

It would be a decent boost (your current i3 is literally the slowest 10th gen i3), but good luck finding a 10th gen CPU for a decent price. The 10600 is currently more expensive on Amazon than more recent and better options like the 12400f. I'm not very well versed on intel socket upgrade compatibility, I know there's a hard cap on 9th gen, but I'm not sure about 10th and up. If a 12th or 13th gen could work in your mobo, it'd be a much better choice than a 10th gen i5.

If you wanna play games, you'd almost certainly be better off saving for a new build.