I would inspect the slot for any debris or visually damaged pins. Also, when you have the card installed check to see if it looks like it is fully seated or if there is more sticking out of the slot on one side.
In order to fully test you probably need to try the card in another motherboard or try a different card in that slot. You could try contacting a local repair shop to assist with diagnosis.
As for the cooling question, there is room for air to move and the power supply should actually pull warm air out of that immediate area. If that is the only slot where the GPU will work then I would just do some testing and monitor temperatures.
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