Actually, they both had short black hair back then but Mizuumi was using the wig that her sister gave her. She probably underwent chemotherapy and had temporarily lost her hair.
The reason Sora had long blond hair during her time as an idol is probably because her style was inspired by Mizuumi's look. This chapter and the previous one suggest that Sora admired Mizuumi as much as Mizuumi admired Sora, if not more.
Yep - Mizu'umi seems to be Sora's direct link & inspiration into becoming an idol in the first place; Sora just so happened to make it much further than Mizu'umi could. Which, makes sense; even as a child, Mizu'umi talks about Sora just having that way about her of actually seeing the people around her and uplifting them. Would make sense that someone like that would be successful in a career like an idol.
The "biggest mistake of my life" didn't make immediate sense because of the way the speech bubbles were oriented - but then I realized it was Mizu'umi who was being discharged, and she convinced Sora to go with her (Against Medical Advice, I figure) - which is what ultimately lead to Sora's death, if I'm putting the pieces together correctly.
Which, would indicate Mizu'umi blames herself for Sora's death, and that would explain her almost fanatical desire to see Sora "live on", even if she has to let Mizu'umi die and become Sora, herself.