Is my daughter's illness and the obstacles she faces in her studies a punishment from Allah for sins I have committed, or does the legal maxim "No bearer of burdens will bear the burden of another" apply here?

Allah does not punish anyone for another's sin in the hereafter. However, in this worldly life, the effects of sins may extend beyond the one who committed them. This is a tribulation, not a punishment for others. Al-Ghazzi said that a person's righteousness rectifies their family, and their corruption corrupts them. Wahb ibn Munabbih said that Allah said: "Beware My wrath, for My wrath reaches three generations, and love My pleasure, for My pleasure reaches the entire nation." He also said: "Verily, when I am obeyed, I am pleased, and when I am pleased, I bestow blessings, and there is no snatching away My blessings. And when I am disobeyed, I become wrathful, and when I become wrathful, I curse, and My curse reaches the seventh generation of offspring."