There would be chaos. Joffrey is a king, so his death would create a power vacuum. His mother Cersei would be furious and would likely start a manhunt for Arya. The Lannisters might declare war on Arya's family or allies. The political landscape would be completely disrupted as different factions would try to take advantage of the situation.
In the fanfic, Arya might kill Joffrey during a big event. She could blend in with the chaos. Since she has great agility and combat skills, she could quickly approach Joffrey while everyone is distracted. For example, during a tournament or a grand feast. She could jump from a balcony or some hidden place, stab him swiftly, and then disappear into the crowd before anyone realizes what has happened.
Yes, there are some out there. You can start by looking on fanfic websites like Archive of Our Own. Just search for 'Joffrey Baratheon gay' and you'll likely find a variety of stories. However, the quality can vary greatly, so be prepared to sift through some to find the ones that are well - written.
There could be a really interesting fanfic where a Dragonborn enters the world of Game of Thrones during Joffrey's reign. Maybe the Dragonborn would initially be seen as a threat by Joffrey. But the Dragonborn's powers could change the power dynamics in King's Landing. For example, if the Dragonborn has the ability to control dragons, it would be a huge advantage over the Lannisters who rely mainly on their wealth and military might at that time.
In the context of the fanfic, the Dragonborn's presence could also expose Joffrey's weaknesses more clearly. The people of Westeros, seeing someone with such great power (the Dragonborn) who is not as cruel as Joffrey, might be more inspired to rise up against Joffrey. And Joffrey's own allies might start to question their loyalty to him when they see the Dragonborn as a potential new power player in the game of thrones.
One common theme is the redemption of Joffrey. OC characters are often used to try and reform his cruel nature. Another theme is power struggle. The OC may get involved in the power dynamics in Joffrey's court. And love or romance can also be a theme, where the OC might develop a romantic relationship with Joffrey or be involved in his love life in some way.
Yes, there are quite a few. You can start by looking on fanfiction platforms like Archive of Our Own. Some writers have created interesting stories exploring the complex relationship between Joffrey and Sansa. However, keep in mind that the nature of their relationship in the original series is rather dark, so these fanfics often delve into complex emotions and situations.