Don't bother overclocking- it doesn't matter when your bottleneck is the graphics card.
For a gaming PC, your graphics card is THE single most important component inside the box, it does 75% of the work in games, not your CPU! In a serious gaming rig, your graphics card is the single component you spend the most on!
$40 cards like the GeForce 210 and Radeon HD 4350 are just HALF the speed of $50 cards like the Radeon HD 4650 and GeForce GT220. And those are 50% slower than $60 cards like the Radeon HD 5570. meanwhile $75-$90 cards like the Radeon HD 5670 and 6670 are another major step up. And for $99 after rebate, the Radeon HD 5770 blows those cards away.
http://www.techspot.com/review/240-ati-radeon-hd-5670/page4.html
No, you won't play much of anything on Intel's integrated GMA X4500. Intel's older GMA options are among the slowest integrated options available, with performance equal to graphics card designs from 8-10 years ago.
http://www.tomshardware.com/reviews/best-graphics-card-radeon-hd-6990-geforce-gtx-590,2912-7.html
You could play Minecraft, Runescape, original Starcraft etc but for any driving or shooter games you'd need some kind of real card. Even the best integrated graphics chipsets are weaker than $35 standalone cards like the Radeon HD 5450. That might make a good temporary card- sell it a month from now, when you upgrade to a gaming card.
Given the relatively low resolution of your TV, the Radeon HD 5670 is a great low-cost choice for your games and it's well below your budget.
http://www.ebuyer.com/product/190138
The best Nvidia card in your price range is the GeForce GTX 550Ti, although it's overkill for a 1366x800 monitor. Even a GTS 450 would run games maxed at that resolution.
http://www.ebuyer.com/product/262659
http://www.ebuyer.com/product/240110
Most games are not heavily multi-threaded, so they run better on fast dual-cores like the E8400 than slower quad-cores like the Q6600 and Q8200.
You've got a Corsair 650W power supply which is sufficient to handle any cards up to the $300 level. Note that for high-end gaming (Crysis, Metro 2033 etc), cards around $120 like the Radeon 5750 and GeForce GTS 450 represent the bottom floor:
http://www.tomshardware.com/reviews/radeon-hd-6790-barts-gpu-geforce-gtx-460,2917-16.html
http://www.techspot.com/review/359-nvidia-geforce-gtx-560ti/page4.html