Not possible for that price range you are looking at the cheapest laptops out pretty much the best you will get with those games is about 30-40 fps on medium with a 500$ laptop. If you can only spend that much your best bet is to build a 500$ desktop
LoL will be able to be playable at 60 FPS. An intel HD 4400 can get 30 - 35 FPS at 1080p with very high settings. I am pretty sure a 840m can get 60 FPS (or pretty close) at 1366x768. Besides, I stated the that Acer is the best option given the $500 budget since it is the only laptop that is currently selling for under $500 with a dedicated graphics chip. I did not state he will get 60 FPS.