Is it permissible, when angered by someone, to strive in worship so that one might be better than them, and is it possible to vent anger internally and say whatever one wishes to oneself until calm is restored before meeting the person?

There is no harm in a Muslim striving diligently in worship to gain reward and recompense, as this is a permissible act affirmed by the Prophet, peace and blessings be upon him. As for speaking ill of, or cursing and reviling, someone one is angry with, this is not permissible because a Muslim's honor is inviolable. The angry person should calm their anger by seeking refuge in Allah from Satan and by performing ablution, as the Prophet, peace and blessings be upon him, said: "Indeed, anger is from Satan, and Satan was created from fire, and fire is extinguished by water. So, if one of you becomes angry, let him perform ablution."