Proof.
Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To prove (1) we have to show that $Rf_*E$ has quasi-coherent cohomology sheaves. The question is local on $S$, hence we may assume $S$ is quasi-compact. Pick $N = N(X, S, f)$ as in Cohomology of Schemes, Lemma 30.4.5. Thus $R^ pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$ and all $p \geq N$ and the same remains true after base change.
First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold for such $E$ with our choice of $N$. In this case we can for example use the spectral sequence
\[ R^ pf_*H^ q(E) \Rightarrow R^{p + q}f_*E \]
(Derived Categories, Lemma 13.21.3), the quasi-coherence of $R^ pf_*H^ q(E)$, and the vanishing of $R^ pf_*H^ q(E)$ for $p \geq N$ to see that (1), (2), and (3) hold in this case.
Next we prove (2) and (3). Say $H^ m(E) = 0$ for $m > 0$. Let $U \subset S$ be affine open. By Cohomology of Schemes, Lemma 30.4.6 and our choice of $N$ we have $H^ p(f^{-1}(U), \mathcal{F}) = 0$ for $p \geq N$ and any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$. Hence we may apply Lemma 36.3.4 to the functor $\Gamma (f^{-1}(U), -)$ to see that
\[ R\Gamma (U, Rf_*E) = R\Gamma (f^{-1}(U), E) \]
has vanishing cohomology in degrees $\geq N$. Since this holds for all $U \subset S$ affine open we conclude that $H^ m(Rf_*E) = 0$ for $m \geq N$.
Next, we prove (1) in the general case. Recall that there is a distinguished triangle
\[ \tau _{\leq -n - 1}E \to E \to \tau _{\geq -n}E \to (\tau _{\leq -n - 1}E)[1] \]
in $D(\mathcal{O}_ X)$, see Derived Categories, Remark 13.12.4. By (2) we see that $Rf_*\tau _{\leq -n - 1}E$ has vanishing cohomology sheaves in degrees $\geq -n + N$. Thus, given an integer $q$ we see that $R^ qf_*E$ is equal to $R^ qf_*\tau _{\geq -n}E$ for some $n$ and the result above applies.
$\square$
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