I am looking at the Pentium G4560 for a new gaming bulid. With this bulid I am looking at playing FPS games and Minecraft/Roblox and brouser games while recording/streaming them.Looking to see if its possible to play these games getting NO lag spikes at max settings while recording these games at 1080p/1440p/4K at 60fps.
simple answer, NO. As Kanewolf has pointed out, that CPU is just too weak to do all the the things you are looking to do. Even a decent 4 core CPU will have a little trouble with streaming, and playing at the res' you've mentioned. You would need a Ryzen5/7 or an I7 4c/8t cpu like a 7700/k.
Streaming doesn't have to be that CPU-dependent, as you can use GPU accelerated encoding. But in this day and age I still wouldn't want to be jumping on a dual core. Especially if you want to be assured of smooth performance at all times.