Besides all your questions, by installing a single stick you hobble the CPU memory speed since it does use the dual channel for better overall memory speed. You're looking at buying RAM and that money may get you a new board and fix the issue.
http://www.amazon.com/Intel-Socket-Motherboard-Retail-BOXDH55PJ/dp/B003K1OVQ0 looks to be 60 bucks and similar but I'd not do what you are proposing.
My PC Specs:
MOBO: Asrock H55M-LE
CPU: Intel Core i3-540 3.06Ghz
RAM: KUK DDR3 1333 4GB
GPU: Nvidia GeForce GTX 650 128BIT DDR5
1st question: I wanted to purchase 8GB RAM alone since my 2nd memory slot is not working and the minimum system requirement of running Call of Duty ghosts is 6GB. I know that this Motherboard supports until 8GB Memory capacity, what my concern is that can my Motherboard supports 8GB RAM alone or do I need to use 4GB x 2 RAM?
2nd question: My Processor supports DDR3 1066/1333 RAM and my Motherboard supports until DDR3 2600 and since I wanted to purchase 8GB DDR3 1600 RAM alone, does that mean that my computer will not run since the Processor only supports until DDR3 1333? Or does that mean that it will run since my Motherboard supports until 2600? Or the computer will run however it will only run at DDR3 1333 speed? I am somewhat doubtful. What will be the basis when you purchase a RAM? Is it the maximum support of your CPU or your Motherboard so that the system will run?