One popular 'Ron Hermione Cursed Child' fanfiction is 'The Unforeseen Bond'. It explores their relationship further after the events of 'The Cursed Child'. It has great character development and interesting plot twists.
You can start by looking on fanfiction websites like Archive of Our Own (AO3) or FanFiction.net. Just type in 'Ron Hermione Cursed Child' in the search bar and a whole list of relevant fanfictions will come up.
A very interesting plotline is when their children have to face a new dark force in the wizarding world. This forces them to rely on the skills and values taught by Ron and Hermione. The story might also include how they use their parents' old friends and resources to combat this threat, while also growing as individuals in their own right.
Well, their child would probably be a very intelligent wizard or witch. Since Hermione is super smart and Ron is also quite clever in his own way. The child might inherit Hermione's love for books and studying, always eager to learn new spells. And from Ron, they could get a great sense of humor and loyalty. Maybe the child would have red hair like Ron and those bright brown eyes like Hermione.
In most cases, reading fanfiction is legal as long as it is for personal enjoyment and not used for commercial purposes without permission. Fanfiction is a form of creative expression by fans, and as long as it doesn't violate any major copyright laws in terms of mass distribution or claiming the work as original, it's okay to read.
One great Ron and Hermione child fanfiction is 'The Next Generation of Magic'. It shows their kids going through Hogwarts adventures, facing challenges just like their parents once did. The character development of their children is really well - done, and it also has some sweet family moments between Ron, Hermione and their kids.
One possible story could be that in these 10 months, Ron and Hermione are on a mission together. They face many challenges that test their relationship. Maybe they have to deal with a dark wizard threat and in the process, their bond deepens as they rely on each other's strengths.
I think there's a good chance the child could be in Gryffindor. Ron is from a long line of Gryffindors, and he has shown great courage throughout his time at Hogwarts. Hermione also has a lot of Gryffindor qualities, like her bravery in the face of danger. So their child would probably follow in their footsteps and be sorted into Gryffindor.
Hogwarts is a magical school, not a sentient being that can read. But if we consider the idea in a more metaphorical sense, within the world of fanfiction, we could imagine that some of the students at Hogwarts might be interested in 'The Cursed Child' fanfiction. They might read it to explore different interpretations of the events and characters they know so well. But in the context of the original series, there's no such indication.