Does it damage the battery when the phone is on the charger, and when I use it like that?

Lithium-Ion batteries in smartphone, tablets and even in a laptop can be charged while using the device. The use itself is not harmful when the device is charging. It is only important that you do not use a charger that is not known by the brand or your smartphone (original charger). The consequence of using a cell phone or tablet while it is charging will be naturally slower battery charge.

Playing games and using similar advanced applications when charging might not the smartest. By heating the device battery is expanding and is generally not desirable that the battery gets too warm (in extreme situations it could ignite). Try to get your smartphone of the high temperatures - do not leave it in direct sunlight or near a heat source. The advantage is that newer batteries are all more resistant to high temperatures.

There were (statistically speaking) very rare cases that the mobile phone (iPhone and even Android smartphone) exploded while using the charger. However, sources say that the cause of the explosion was unknown brand charger that was used instead of the original one.

Preferably, you can leave the smartphone and the battery "rest" every once a week. This means that you need to turn it off every now an then for a few hours (overnight for example). Also, do not wait until the battery drains out - it is better to charge it every now and then.