MacBook – Should I be concerned that the fan on the 11 inch Air makes a lot of noise when I play Call of Duty Black Ops

Yes, it could. Some things that could be happening, in order of probability:

1. Your computer's innards could be dusty. This is the the most likely cause, and is easily remedied by unplugging your computer, taking it outside or into the garage, and using a compressed air canister to blow out all of the dust in the vents. (If you have allergies or asthma, I suggest wearing safety goggles and/or a mask. Also, the can will get very cold, so wearing gloves is probably a good idea as well.)
2. A(n unrelated) background process could be taking up a lot of CPU. What was the CPU percentage? Did you check if any other processes were running? If this is happening often, it could be that some other program (Software Update, Finder, Skype, etc.) is trying to run in the background, and is experiencing errors that cause it to hog memory.
3. The game and/or media player you use may have recently undergone an update that consumes significantly more CPU and/or memory and/or disk usage that you are used to. If you ran a software update in the last two or three weeks, or if the game and/or media player performs automatic updates, this may explain the problem. (If this is the case, and the issue is very bothersome, you can probably contact the vendor to get the previous version.)
4. A government entity could be spying on you through your computer.

"Fully charged" does not mean exactly 100%, but somewhere between 95% (or even less) and 100%. Obviously it is better for battery life if the battery is not immediately loaded after a small percentage of charge was consumed, if your MacBook is connected to the charger. So you do not have an issue with your battery.