While there may be some of wanting to "toughen up" Futaba in her motivation, or at the very least it might be how she's justifying it to herself, I don't think it's a matter of pure intentions and horrible execution. As mentioned, she had to survive among the Yakuza. She probably learned early that centring out someone else is a good way to avoid being targeted yourself. It's probably at least partially a survival tactic on her part. Furthermore, Futaba's got both parents, and while they're overseas, they still regularly send her money, and she obviously has a loving relationship with them. There might also be some jealousy thrown into the mix.
I don't think Yumi is a bad person at heart. I think she's broken, like pretty much everyone in the story (still waiting to see how some of the people are broken). I like her, when looked at on the whole, as a character for the VN, and look forward to seeing more of her. However, I think there's more to her bullying than simply trying to toughen Futaba up.
We don't really know what happened in the previous detentions. It's possible previous Senseis were as bad or worse than this one, and they could've been unjustified detentions, and were excuses to abuse her. It's equally possible that they were justified and the other Sensei was kind and dedicated to helping her. Everything anyone says about them at this point, beyond the fact that they happened, and Yumi didn't like them (and really, who likes detention?), is pure speculation.