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On Q3, maybe an additional geometric remark. First, to take the non-torsion field $k$ as coefficients, you should use the pro-étale topology, not the étale one, as Scholze does. Moreover, 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 at every point, contrary to the complex analytic case, but a rational domain of the affine space, or of a disk (take for example the Gauss point of the circle, whose neighborhoods are annuli). 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, so that one may prove the local Poincaré's lemma as in the case of smooth manifolds (there is a nice mathoverflow answer on this by David Speyer).

However, every point of the rigid analytic space in the classical sense (with residue field finite over the base field) has a fundamental system of étale neighborhoods given by polydiscs, on which the Poincaré lemma is true, so that one may show that the de Rham complex is locally exact at these classical points for the Berkovich étale topology. This is discussed in details in Berkovich's paper ``Integration of one forms on $p$-adic analytic spaces''.

There is also 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 the (pro-)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).

Another Poincaré lemma in the $p$-adic setting based on the $h$-topology and derived de Rham cohomology is due to Beilinson (see also Bhatt's work).

On Q3, maybe an additional geometric remark. First, to take the non-torsion field $k$ as coefficients, you should use the pro-étale topology, not the étale one, as Scholze does (see Scholze and Bhatt's paper on pro-étale cohomology). This may correct partly the well identified problem with Galois cohomology that appears in the calculation of Jon Brener's answer. Moreover, 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 at every point, contrary to the complex analytic case, but a rational domain of the affine space, or of a disk (take for example the Gauss point of the circle, whose neighborhoods are annuli). 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, so that one may prove the local Poincaré's lemma as in the case of smooth manifolds (there is a nice mathoverflow answer on this by David Speyer).

However, every point of the rigid analytic space in the classical sense (with residue field finite over the base field) has a fundamental system of étale neighborhoods given by polydiscs, on which the Poincaré lemma is true, so that one may show that the de Rham complex is locally exact at these classical points for the Berkovich étale topology. This is discussed in details in Berkovich's paper ``Integration of one forms on $p$-adic analytic spaces''.

There is also 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 the (pro-)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).

Another Poincaré lemma in the $p$-adic setting based on the $h$-topology and derived de Rham cohomology is due to Beilinson (see also Bhatt's work).

On Q3, maybe an additional geometric remark. First, to take the non-torsion field $k$ as coefficients, you should use the pro-étale topology, not the étale one, as Scholze does. Moreover, 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 at every point, 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, so that one may prove the local Poincaré's lemma as in the case of smooth manifolds (there is a nice mathoverflow answer on this by David Speyer).

However, every point of the rigid analytic space in the classical sense (with residue field finite over the base field) has a fundamental system of étale neighborhoods given by polydiscs, on which the Poincaré lemma is true, so that one may show that the de Rham complex is locally exact at these classical points for the Berkovich étale topology. This is discussed in details in Berkovich's paper ``Integration of one forms on $p$-adic analytic spaces''.

There is also 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).

On Q3, maybe an additional geometric remark. First, to take the non-torsion field $k$ as coefficients, you should use the pro-étale topology, not the étale one, as Scholze does. Moreover, 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 at every point, contrary to the complex analytic case, but a rational domain of the affine space, or of a disk (take for example the Gauss point of the circle, whose neighborhoods are annuli). 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, so that one may prove the local Poincaré's lemma as in the case of smooth manifolds (there is a nice mathoverflow answer on this by David Speyer).

However, every point of the rigid analytic space in the classical sense (with residue field finite over the base field) has a fundamental system of étale neighborhoods given by polydiscs, on which the Poincaré lemma is true, so that one may show that the de Rham complex is locally exact at these classical points for the Berkovich étale topology. This is discussed in details in Berkovich's paper ``Integration of one forms on $p$-adic analytic spaces''.

There is also 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 the (pro-)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).

Another Poincaré lemma in the $p$-adic setting based on the $h$-topology and derived de Rham cohomology is due to Beilinson (see also Bhatt's work).

On Q3, maybe an additional geometric remark. First, to take the non-torsion field $k$ as coefficients, you should use the pro-étale topology, not the étale one, as Scholze does. Moreover, 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, so that one may prove the local Poincaré's lemma as in the case of smooth manifolds (there is a nice mathoverflow answer on this by David Speyer).

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).

On Q3, maybe an additional geometric remark. First, to take the non-torsion field $k$ as coefficients, you should use the pro-étale topology, not the étale one, as Scholze does. Moreover, 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 at every point, 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, so that one may prove the local Poincaré's lemma as in the case of smooth manifolds (there is a nice mathoverflow answer on this by David Speyer).

However, every point of the rigid analytic space in the classical sense (with residue field finite over the base field) has a fundamental system of étale neighborhoods given by polydiscs, on which the Poincaré lemma is true, so that one may show that the de Rham complex is locally exact at these classical points for the Berkovich étale topology. This is discussed in details in Berkovich's paper ``Integration of one forms on $p$-adic analytic spaces''.

There is also 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).