Let $X_0$ be a smooth variety (for simplicity I'm willing to assume that X is a curve) over a finite field $k$, $X$ its geometric base change, and $\mathcal{F}$ an $l$-adic etale sheaf on $X$ with $\mathcal{F}(m)$ its Tate-twist.

What is the relation between the weights of Frobenius on $H^r(X,\mathcal{F})$ and $H^r(X,\mathcal{F}(m))$?

I remember seeing somewhere (in Milne?) that $H^r(X,\mathcal{F})\otimes \mathbb{Z}_l(m)\cong H^r(X,\mathcal{F}(m))$ under some restriction to $k$, but I can't find it now. Also, it strikes me as an odd result, considering that the analogue for Serre twists in the classical sheaf cohomology setting doesn't hold.

While searching for an answer in Milne, Etale Cohomology i read about the concept of Purity. Say, I'm willing to assume $\mathcal{F}$ is pure of some weight.

Is there any relation between "Cohomological Purity" in the sense of Milne, namely that something happens in a certain codimension, and the concept of a sheaf being of "pure weight" $s$, i.e. having Frobenius eigenvalues with complex norm $|k|^{s/2}$?