I've thought about it and that I could go either way since I'll be able to develop her better. What do you think?

I'm fine with Yusuke and Botan handling the first case for the most part. And I do consider Kuwabara and Yusuke rivals but I also consider them friends, considering how much Yusuke helped Kuwabara with his test, when he was still dead. They still get to insult each other though I see it more as something between friends, as opposed to real rivals. Hiei and Yusuke, seem more like true rivals, at least power wise. But I'm fine with them starting off with some misunderstood animosity though I'd like to explain it a bit since the context prior to the series, to me felt unclear. I know Yusuke and Kuwabara were bullies of sorts, or thought as such but I never did understand why they were so against one another. Did you know? And I would say it's logical to assume Keiko and Yusuke did grow apart since the high school years would the ones where you branch out, meet new people, discover what you like and what you might want to do after high school.

Middle/winter sounds fine to me. I think most schools are "year round" in Japan so they have a lot of breaks but nothing as extended as America's summer vacation. Though I do think they would have maybe 1-2 weeks off for the new year and other holidays so that would give them some good time to either finish the case or subdue it. We could start with a few days before the break with Botan seeking Yusuke out at school so we're not having a case starting "right" when he's off of school, but leading into it?