Arya and Tywin could also interact in a more unexpected way. Maybe in a situation where they are forced to work together for a short time due to some external threat. During this time, Arya has to constantly remind herself of her ultimate goal of revenge while Tywin is constantly on guard against her, yet also seeing potential in her skills for his own purposes.
Joffrey was a cruel character. It's hard to imagine a traditional love story for him. His nature was so malicious that any form of love seemed out of place. He was more interested in power and causing pain. So, in the context of the show, a real love story for him seems almost impossible.
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.
There could be a fanfiction that focuses on an alternate universe where Tywin survives longer. In this story, he would have to deal with the changing political landscape after the events at King's Landing. He might try to regain control of the situation, perhaps by making alliances with unexpected parties. His character would be forced to adapt to new threats and challenges, which could lead to some very interesting plot developments.
One interesting Tywin fanfiction could be one that delves into his early years and how he rose to power. It might explore his relationships with his father and the events that shaped his cold and calculating nature.
One great 'Game of Thrones Tywin Lannister Fanfiction' is 'The Lion's Legacy'. It delves deep into Tywin's past, showing his rise to power in a very detailed and engaging way. The author does an excellent job of portraying his complex personality.
One possible fanfiction could be about Joffrey's brother Tommen growing into a wise and just ruler. He could be shown learning from the mistakes of his older brother. He might start secretly training in combat and political strategy, with the help of some of the more noble characters in the series, like Tyrion perhaps. He could then start to make changes in King's Landing, trying to reverse the chaos that Joffrey had caused.
Joffrey is often characterized as even more cruel and unhinged in fanfiction. His violent outbursts and lack of mercy are sometimes exaggerated. Margaery is usually shown as a very intelligent and manipulative woman. She uses her beauty and charm to her advantage.