Wow, do I even want to interject here?
In which case this begs a question... why suggest him buy A4-6300 if he will have to buy a Graphics Card anyway?
Its a 40 dollar CPU, you get what you pay for. Its an APU just like ALL of Intel's CPUs have been for years. Under your logic I could say "why buy an i7 when you're just going to buy a graphics card anyway"? Good lord dude, cool your jets.
And why not buy some actual half-decent CPU without integrated GPU crap AND a graphics card?
Because you pretty much can't avoid integrated "GPU crap" anymore as I said already. Both major CPU manufacturers have clearly demonstrated this continued integrating feature as being the industry standard.
Now to get to the OPs question, no its not an ideal CPU for "gaming", I'm not sure what this one guy's deal is where hes ranting and raving because a 40 dollar CPU doesn't perform like a 200 dollar one (possibly "mommy issues", but who can say?), but the old adage you get what you pay for does apply here. I run the 6300 in an HTPC that doubles as a Minecraft homebox, it does that fine, would I try to power up Battlefield 4 on ultra on it? No...
The call of duty games are not particularly demanding at all, you wouldn't have much trouble running them with an A4 however. The other titles you mentioned I have no first hand knowledge to report, but I ran COD MW 3 on an old laptop with a Core2Duo just fine.
As far as overclocking at the BLCK, unless AMD has radically changed their design (which I doubt), its fine with their CPUs. When you overclock at the multiplier you are only overclocking the CPU. At the base clock, you are also overclocking your RAM and your North Bridge on an AMD CPU. So, you're introducing more potential stability trouble points. However, its fine to do so, as long as your temps are good and your system is stable. Now with Intel you don't want to overclock at the base clock, as they have literally tied everything into it, including the PCI lanes, SATA even the ethernet. Some of those things even an adjustment of 1 or 2mhz can cause severe stability problems.
As far as the stock AMD cooler, its adequate from first hand experience. For overclocking? They're not, but for overclocking, Intel's stock heatsinks are equally useless.