First check your Power settings ( right click on the battery icon and press power options)
by default when you are plugged in to AC it'll be in High Performance and will be switched to Power saver or Balanced when you unplug. Change the settings to your need and will get smooth performance.
it is OK to play games while plugged in and battery is fully charged. i does it all the time.
The higher end gaming laptops draw more power than the battery can deliver alone. My i7 quad + 970m laptop needs about 150 Watts while its battery can only provide about 100 Watts. So it degrades the GPU in battery-only mode to keep total power draw under the 100 Watt limit. It has to be plugged into AC for the GPU to run at full speed.
Some of the higher-end gaming laptops draw more power than even the AC adapter can provide. Those will slowly drain your battery when you're playing while plugged in. If you remove the battery (or drain it entirely) and play on just AC power, the GPU will again run in a degraded mode.
And yes it's fine to play while plugged in and the battery fully charged.