When you unplug a laptop it should automatically go into a power saving mode in order to extend battery life.
When it is plugged in it will run at full power.
If you run full power with it unplugged and play games or watch movies you will drain the battery much faster.
So if you are playing games, do so with the laptop plugged in if you can. Games should perform better and you wont be limited to how long you can play due to battery life.
No it shouldn't affect the battery's health. They are going to wear out eventually, and since yours is built in it should be a high quality battery. Though it just makes it a pain to deal with should you ever have to have it replaced.