What is the ruling on a teacher uttering obscene words, and cursing and insulting students, even though he is observant of prayer, Qur'an recitation, and fasting? And what is the ruling on a Muslim cursing his brother jokingly, or laughing when subjected to cursing?

What the teacher does, using obscene language, cursing, and insulting others jokingly, is not appropriate. Indecency and vulgarity are dispraised in Islamic law. As for your laughter when someone curses you, you are not sinful for it. However, you should advise the one who curses and explain the ruling to him.