On Q3, maybe an additional remark. I think the point is that, even locally for the étale analytic topology (either in Huber's quasi-étale sense or in Berkovich's sense), a smooth rigid space is not a disk, contrary to the complex analytic case, but a rational domain of the affine space, or of a disk. Such a rational domain may be non-contractible in the de Rham sense, i.e., have non-trivial higher de Rham cohomology, so one can't show that the de Rham complex is locally exact, even for the analytic étale cohomology, in higher degrees. This is to be compared with the algebraic situation, where the Zariski topology is not fine enough to make this complex locally exact, and to the complex analytic situation, where manifolds are locally given by polydiscs, that are ``de Rham contractible'' in the above sense.
There is a $p$-adic Poincaré lemma for smooth proper rigid analytic varieties, due to Scholze (corollary 6.3 of ``$p$-adic Hodge theory for rigid analytic varieties''), that uses the pro-étale topology on $X$, and a version of the de Rham complex defined using a sheaf of periods $B^+_{dR}$ as an analog of the constant sheaf. This implies the isomorphism between étale cohomology with coefficients in $B_{dR}$ and de Rham cohomology of $X_{B_{dR}}$, that was previously known in the semi-stable algebraic situation, by Faltings and others (by other methods).