Stock Cooler doesn’t hurt unless you overclock.
You will need to change it(cooler) in a year or two for bit better performance.
For light gaming and tight budget processor is not bad. But try ryzen or wait for ryzen 3000?
You probably cannot overcock i3.
Gaming aside, programs like AutoCad and Adobe CC work far better on Intel than they do Ryzen. So even with the fact you are looking at an i3, it's still a better choice than the 2200G. Either cpu is upgradable later