One reason could be betrayal. If someone he trusted, like a fellow ninja or an old acquaintance, turned out to be working for his enemies, that would surely make him pissed. It's a huge blow to his trust, which is a big part of his character.
If Raditz survived in fanfiction, he would likely have a complex relationship with Goku. He might start to see the value in Goku's way of life. Goku could try to reform him, and Raditz could slowly become a more positive force in the Dragon Ball Z universe.
Sasuke could be a possible cause. If Sasuke were to go back on his word about protecting the village or if he were to harm their friends again, Naruto would be pissed. Their relationship is complex, and Sasuke's actions often have a strong impact on Naruto.
Sasuke could make Naruto pissed off if he goes back to his old ways of seeking power and revenge. Naruto has worked hard to bring Sasuke back to the right path, and if Sasuke strays again, it would surely anger Naruto.
Raditz could be redeemed by sacrificing himself for the greater good. For example, if there was a powerful enemy threatening the Earth and he steps in to take a fatal blow meant for someone else, like Gohan or Goku.
A new, powerful and arrogant villain introduced in the fanfic could be the cause. This villain could constantly mock Naruto and his ideals, and try to undermine his efforts to protect the village. Such blatant disrespect would make Naruto really pissed off as he is very protective of his home and the values he stands for.
Sasuke could be the cause. If Sasuke were to betray Naruto again in the fanfiction, it would surely make Naruto pissed off. Their relationship is complex and Sasuke's actions have a big impact on Naruto.
No, these are not common. People usually try to maintain their dignity and privacy, and such situations are not things that are regularly talked about in general society. It involves personal privacy and bodily functions which are not topics for normal public conversation.
A good Raditz story should focus on his Saiyan heritage. For example, showing how his Saiyan instincts play out in different situations, like when he's facing a tough enemy or making a decision about whether to show mercy or not.