The Sea of Japan or 동해 as it’s known in Korea (East Sea) are the same thing. The two names will be used interchangeably in my posts that involve the East Sea. I will probably refer to it as East Sea more often (it’s shorter) but I have no real strong opinion one way or the other.
독도 (dokdo) or Takeshima are both the same group of islets. They are sometimes known as Liancourt Rocks. Dokdo is something I have an opinion on (one I am making sure is completely informed) and I will always refer to them by their Korean name until I have reasonable doubt they should be referred to as something else or their status changes officially. However, as it stands Korea is the administrator of the islets so I will use their Korean name.