No: this fails already when $X=E$ is an elliptic curve and $k=\mathbb{Q}$. This would imply that every element of $H^1(\mathbb{Q},E)$ has order at most $d$, and I'm pretty sure that this cohomology group has elements of arbitrarily large order.

Your question is closely related to whether the order of every element of $H^1(k, \mathrm{Aut}(\overline{X}))$ is bounded. It is not in general, but is in many cases.

No: this fails already when $X=E$ is an elliptic curve and $k=\mathbb{Q}$. This would imply that every element of $H^1(\mathbb{Q},E)$ has order at most $d$, and I'm pretty sure that this cohomology group has elements of arbitrarily large order. Otherwise the Tate-Shafarevich conjecture would be rather trivial, as it is well-known that $Sha(E)[d]$ is finite for every $d$.