No, in the real 'Game of Thrones' story Arya didn't kill Joffrey. Joffrey was poisoned at his own wedding. But fanfic can be really wild. Some fanfic writers might write a scenario where Arya somehow manages to kill Joffrey. Maybe they write it because Arya has a long list of people she wants to kill and Joffrey is a very hated character. But it's not part of the original plot.
In the official 'Game of Thrones' series, it was Sansa's necklace (which was actually a poisoned device) that led to Joffrey's death, not Arya. However, in some fanfics, authors might create alternate storylines where Arya kills Joffrey. But these are non - canonical and just creative interpretations by fans.
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.
If Arya killed Joffrey in a fanfic, one possibility is that she would become an instant target for the Lannisters. They would stop at nothing to avenge Joffrey's death. Arya would have to go into deeper hiding or rely on her Faceless Men training even more. Maybe she'd have to flee across the Narrow Sea and start a new life in Essos, constantly looking over her shoulder for Lannister assassins.
In some fanfics, Arya Stark might be presented as the last Stark, but it really depends on the specific plot and direction the fanfic author takes. There could be plot twists where other Starks are revealed to still be alive.
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.
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.
There are some really interesting fanfics out there that combine the concepts of a Dragonborn in the Game of Thrones universe with Joffrey. One I've read had the Dragonborn as a sort of mysterious figure who arrives in King's Landing during Joffrey's reign. The Dragonborn's presence throws the power dynamics into chaos as they have their own set of powers and motives. It was really fascinating to see how the author played with the different personalities and politics of the Game of Thrones world in that fanfic.
One idea could be a story where Arya meets Lemon in a tavern during her travels. Maybe Lemon has some interesting information about Arya's enemies, and they form an unlikely alliance to take on a common foe.
A fanfic involving a Dragonborn in the Game of Thrones universe with Joffrey could be a wild ride. Maybe the Dragonborn, with their dragon - riding abilities and magic, would pose a significant threat to Joffrey's tyrannical rule. They could swoop in on their dragon, causing chaos in King's Landing and challenging Joffrey's claim to the throne.
Fanfic writers might explore Arya's warging in different ways. They could show how she discovers this ability on her own, or perhaps she meets someone who helps her understand and control it. It could also be used to create new and exciting plotlines, like her warging into a bird and spying on the Lannisters or other powerful families in King's Landing.