Is $X$ connected? What do you mean by "quasi-split" in the relative setting? (Just existence of a Borel $X$-subgroup $B$, or also $B$ contains a fiberwise maximal $X$-torus, or...? SGA3, XXIV, 3.9 demands a further condition in terms of the scheme of Dynkin diagrams, automatic for semi-local $X$.) Identifying ${\rm{H}}^1(X',\mathbf{G})$ with ${\rm{H}}^1(X,{\rm{R}}_{X'/X}(\mathbf{G}))$ and taking inspiration from Prop. 36 in section 5.4 of Ch. I of Serre's "Galois cohomology" book, the set of such isom. classes is $\mathbf{G}(X')\backslash ({\rm{R}}_{X'/X}(\mathbf{G})/G)(X)$.

@AndreasHolmstrom: I agree 100% with your comment to Qiaochu. But note that in your elliptic curve example you are strictly speaking dropping a couple of Riemann-zeta factors that corresponding to degree-0 and degree-2 cohomology on fibers; i.e., you are really speaking about the degree-1 part, which is not the entire "zeta function". Curves are misleading in that a single cohomological part can still be expressed via point-counting; in general one cannot expect this to happen. My answer below discusses it in more detail.

Just to be clear, in the definition of the category $\mathcal{D}$, "complete" is not meant in the sense of max-adic topologies (i.e., it is not something determined by the underlying local ring), but rather must be meant in the sense of pseudo-compact rings. For your setup with finite residue fields, it means that the rings are equipped with a profinite topology (which is however not generally the topology of finite-index additive subgroups), and morphisms are required to be continuous (so not just a locality condition determined by the maximal ideals).

@LiYutong: In the case of interest one can see that the degree of the $f_t$'s is bounded, so that can avoid your proposed counterexamples (if we allow ourselves to drop some $t$'s). But degree bound alone is insufficient, as my previous comment indicates.

It isn't true that any two pointed finite maps from $C$ onto a fixed target curve are related through an automorphism of the source: at the very least one needs to bring in degree bounds (which certainly hold in the case of interest) but one also has to rule out other things such as automorphisms of the target which preserve the base point (of which there can be infinitely many when the target curve is a rational curve).

@MichaelZieve: The field you call $L$ isn't algebraic over $K$, due to the intervention of formal power series. I think you mean $L$ to be the maximal unramified extension of $K$, which is to say the direct limit of the finite extensions $\mathbf{F}_{q^n}(\!(t)\!) = \mathbf{F}_{q^n} \otimes_{\mathbf{F}_q} K$ of $K$.

@S.Carnahan: The mistakes in the original version of this answer have been fixed.

See Chapter XIII, section 3--6 of Lang's "Real and Functional Analysis" for a very elegant treatment under which higher derivatives are defined via multilinear maps and moreover the $p$th higher derivative is genuinely the "derivative" of the $(p-1)$th. One virtue of this approach is that it permits both a formulation and proof of the higher-dimensional Taylor formula which looks and feels exactly like the 1-dimensional case (with the mess of factorials hidden away within a clean formalism); of course, one can bust out coordinates and recover the usual messier explicit version from that.

By the way, since PGL$_n$ represents the automorphism functor of projective $(n-1)$-space (by deformation theory to bootstrap from the classical case on field-valued points), failure of surjectivity is exactly the condition of $\mathbf{P}^{n-1}_R$ admitting an $R$-automorphism which does not arise from an invertible $n \times n$ matrix over $R$.

The diagram $1\rightarrow{\rm{GL}}_1\rightarrow {\rm{GL}}_n\rightarrow{\rm{PGL}}_n\rightarrow 1$ of smooth affine $R$-group schemes is exact for the etale topology, so the obstruction to surjectivity on $R$-points is the triviality of the induced map between the first two etale-topology H$^1$'s, which by descent theory is the map ${\rm{Pic}}(R)\rightarrow {\rm{Vec}}_n(R)$ carrying a line bundle $L$ to $L^{\oplus n}$. So short-exactness on $R$-points is equivalent to $L^{\oplus n}\simeq R^n$ iff $L\simeq R$. Considering det, this holds if ${\rm{Pic}}(R)[n]=0$ and conversely for Dedekind $R$.

I required dimension $> 1$ precisely because group actions on smooth curves exhibit much simpler behavior than in higher dimensions (e.g., passing to coarse quotient preserves smoothness, unlike nearly all cases beyond dimension 1, and the stack quotient is certainly smooth). Do I misunderstand your example? But (1) is true in great generality since finite group invariants preserves normality (I'm not sure why you think it fails). The failure of log-smoothness should occur beyond dimension 1 in char. 0 when $G$ is highly non-abelian, but for a rigorous proof ask a local expert in log geometry.