I have a flat, quasicompact, and separated map $p : X \to \mathbb{A}^1$ and I know that $R^i p_* \mathcal{O}_X$ vanishes everywhere except possibly $0 \in \mathbb{A}^1$.
Q1: Does "cohomology and base change" automatically hold here? I.e., is the natural map $(R^i p_* \mathcal{O}_X)_s \otimes_{\mathcal{O}_{\mathbb{A}^1, s}} k(s) \to H^i(X_s, \mathcal{O}_{X, s})$ an isomorphism for all $s \in \mathbb{A}^1$? This is clear for all nonzero $s$ and I believe it should follow for $0 \in \mathbb{A}^1$ automatically.
Q2: Does it then follow that $R^i p_* \mathcal{O}_X = 0$? Theorem 1.1 in Conrad's notes would then imply that $R^i p_* \mathcal{O}_X$ was locally free, hence constant rank 0, hence 0.
Q3: If $X \overset{v}{\to} Y \to \mathbb{A}^1$ are proper maps and both $X$ and $Y$ are flat over $\mathbb{A}^1$ ($v$ is proper birational, n.n. flat), then does cohomology of $\mathcal{O}_X$ along $v$ commute with base change in $\mathbb{A}^1$? The versions of Cohomology and Base Change I've seen have one map which must be both proper and flat, as opposed to a composite being flat and the first map proper. This is my original situation, and I got to the above by localizing in $Y$.
I've been trying to do this descending induction, where $R^N p_* \mathcal{O}_X$ is zero for $N \gg 0$ for dimensional reasons, so $R^{N-1} p_* \mathcal{O}_X$ commutes with base change, but I also need $R^{N-2} p_* \mathcal{O}_X$ to commute with base change to conclude $R^{N-1}p_* \mathcal{O}_X$ was locally free, hence zero, and continue.
