= Solution
Take an affine open subscheme $U=\operatorname{Spec}A\subseteq Y$. Properness and flatness survive base change, and $A$ is reduced because $Y$ is reduced. A bounded complex of finite locally free modules computes the cohomology of $\mathcal F$ on $X_U$.
If some $H^p(X_U,\mathcal F)$ were nonzero, choose the largest such $p=n$. All groups above degree $n$ would vanish, while every fiber group in degree $n$ vanishes by hypothesis. Part (iii) would force $H^n(X_U,\mathcal F)=0$, a contradiction. Hence
$$
H^p(X_U,\mathcal F)=0
$$
for every affine $U\subseteq Y$ and every $p\ge0$. These groups compute the sections of the higher direct images over affine opens, so $R^pf_*\mathcal F=0$ for all $p$, including $f_*\mathcal F=0$ when $p=0$. The <Leray spectral sequence> now gives
$$
\boxed{H^p(X,\mathcal F)=0\quad(p\ge0).}
$$
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