One common theme is redemption. Maybe Odin is trying to make up for past wrongs towards Loki. Another is family drama, as their relationship is complex in the original mythology.
Yes, there are quite a few. You can start by looking on popular fanfiction websites like Archive of Our Own. There, you'll find various takes on the 'odin loves loki' concept. Some might explore it in a more platonic way, while others could be more on the romantic side.
Loki's character could develop in terms of his understanding of power. Through the abuse, he realizes that power can be misused like Odin does. So he may strive to use his own power in a more just way. He might also start to question the Asgardian way of ruling which is based on Odin's actions.
One possible storyline could be Loki's struggle to break free from Odin's abuse and find his true self. He might start to gather allies secretly, like some of the more sympathetic Asgardians. Another could be Loki's internal journey of dealing with the emotional trauma, perhaps having flashbacks of the abuse while trying to perform his duties in Asgard.
A theme of acceptance is also prevalent. Loki is often seen as an outcast, and Odin's love can represent acceptance of Loki's true self, whether it's his magical abilities, his different nature compared to the other Asgardians, or his past mistakes. This acceptance can be a slow and arduous process in the fics, full of emotional ups and downs.
One great 'odin loves loki fanfic' is 'A Father's Unspoken Love' on FanFiction.net. It delves deep into Odin's complex feelings for Loki, exploring their relationship from a more emotional and psychological perspective.