You must be reading the Beauville--Laszlo paper "Un lemma de descente". Go beyond the "abridged English version" at the start of the paper and read the more detailed French version (= the actual paper). There you'll find Lemme 3(a) which answers your vanishing question. For your other question, a trivial example is when $A = \widehat{A}$ (but not noetherian), such as $A = R[\![z]\!]$ with $f = z$ and $R$ not noetherian. Slightly less trivial example: take $A$ to be the integral closure of a discrete valuation ring $A_0$ in an algebraic closure of its fraction field, $f$ a uniformizer of $A_0$.

Unraveling definition, the set of $\mathbf{C}$-points of ${\rm{R}}_{\mathbf{C}/\mathbf{R}}(X_{\mathbf{C}})$ is $X(\mathbf{C}\otimes_{\mathbf{R}}\mathbf{C}) = X(\mathbf{C})\timesX(\mathbf{C})$ via the $\mathbf{C}$-algebra isomorphism $\mathbf{C}\otimes_{\mathbf{R}}\mathbf{C}=\mathbf{C} \times\mathbf{C}$ via $a\otimes b\mapsto (ab,a\overline{b})$. If you replace $X_{\mathbf{C}}$ with a general affine algebraic $\mathbf{C}$-scheme $Z$ inside the Weil restriction you get $Z(\mathbf{C})\times \overline{Z}(\mathbf{C})$ where $\overline{Z}$ is the twist of $Z$ by complex conjugation on $\mathbf{C}$.

Hmm, I seem to be miscalculating a lot today. The rationalization of the group of $\chi$'s seem to be controlled (in the sense of a short exact sequence) by two copies of the character group: one copy that keeps track of unramified characters and the other which keeps track of ${\rm{Lie}}(\chi)$. Probably a combination of the above "answer" and your argument in the unramified case should then settle the general case up to torsion characters. Please let me know if that leads to some progress. (The torsion aspect sounds like it could be delicate.)

Isn't the unramified $\chi({\rm{diag}}(x,y)) = (-1)^{{\rm{ord}}_p(x/y)}$ an unramified counterexample of GL$_2$? It seems likely that torsion $\chi$ will generally create problems, and that is where the intuition related to Weyl chambers breaks down.

Please say that $T$ is assumed to be split (as split $G$ have plenty of non-split maximal $k$-tori when $k \ne k_s$) and that the root system you're speaking about is the one associated to $(G,T)$. Also, please say that Inn($G$) means $G^{\rm{ad}}(k)$ (as opposed to $G(k)/Z_G(k)$) and say whether Inn$(G,\theta)$ means the $k$-points of the identity component of the $\theta$-centralizer in $G^{\rm{ad}}$ or the group of all $k$-points of the $\theta$-centralizer in $G^{\rm{ad}}$ (a priori the latter could be bigger).

A beautiful generalization is given as Lemma 1.19 in Ch. II of the paper of Deligne and Rapoport on generalized elliptic curves (and proved via valuative criterion for properness and Zariski's Main Theorem): if $f:X \rightarrow Y$ is a separated quasi-finite flat map of finite presentation and the fiber-rank function is locally constant on $Y$ then $f$ is finite. (They assume $X$ and $Y$ are noetherian, but the proof can be adapted to work without that condition.)

You use "real algebraic subvariety" and "real algebraic action" with non-standard meaning; please define your intent with such terms. If $H$ is a split connected reductive $\mathbf{R}$-group (e.g. SL$_2$), $X={\rm{Bor}}_{H/\mathbf{R}}$ is the $\mathbf{R}$-variety of Borel subgroups, and $x\in X(\mathbf{R})$ then $H$ acts on the Weil restriction $X':={\rm{R}}_{\mathbf{C}/\mathbf{R}}(X_{\mathbf{C}})$ with orbit through $x$ exactly $H/B_x\subset X'$ and $(H/B_x)(\mathbf{R})=H(\mathbf{R})/B_x(\mathbf{R})$ by Hilbert 90. So it is $\mathbf{R}$-points of something $\mathbf{R}$-algebraic in $X'$.

Typo in above comment: in the final sum indexed by characters $\chi \in \Delta - I$, should be summing the associated (isogeny category) "cocharacters" $\chi^{\ast}$ in the dual basis $\Delta^{\ast}$.

For any $k$ of char. 2, $B_q$ on the hyperplane $H=v^{\perp}$ has 1-dimensional defect space $L$, with symplectic form $\overline{B}$ on $H/L$. Also, $G:={\rm{Stab}}_v(O(q))$ has 2 connected components, each geometrically connected over $k$, and $G^0 \rightarrow{\rm{Sp}}(\overline{B})\simeq{\rm{Sp}}_{2n-2}$ is a $k$-group isomorphism. Indeed, WLOG $k = \overline{k}$, so WLOG $q(v)=1$, and then use the self-contained proof of Prop. C.3.1 of math.stanford.edu/~conrad/papers/luminysga3.pdf. For finite $k$ this proof gives $G(k)={\rm{Sp}}_{2n-2}(k)\times\mathbf{Z}/(2)$ by Lang's theorem.

@Arkandias: You seek $\lambda$ without specifying $M$ (so with $P_G(\lambda)=P$ you could choose $Z_G(\lambda)$ to be $M$). Let $B$ be a minimal parabolic $k$-subgroup of $G$ contained in $P$ and containing $S$, so $\Phi(B,S)$ is a positive system of roots in $\Phi(G,S)$ contained in the parabolic set of roots $\Phi(P,S)$. Let $\Delta$ be the base of $\Phi(B,S)$, $\Delta^{\ast}$ the dual basis of X$_{\ast}(S\cap\mathscr{D}(G))_{\mathbf{Q}}$. Then $P=P_I$ with $I\subset\Delta$ as usual. Use $\lambda=N\cdot \sum_{\chi \not\in I}\chi$ for $N>0$ divisible to make $\lambda \in{\rm{X}}_{\ast}(S)$.

Presumably you mean for $X$ and $Y$ to be of finite type over $k$, but even when $k$-smooth it isn't true: consider ${\rm{Spec}}(k') \rightarrow {\rm{Spec}}(k)$ for a nontrivial finite extension $k'/k$ (separable to ensure smoothness, say), as the tangent spaces are then 0. But if also isomorphism between residue fields at closed points then it suffices that $Y$ is smooth and $X$ is either smooth or geometrically connected over $k$. This is an exercise with etale morphisms.

The document to which Dimitrov refers is at this link: math.stanford.edu/~conrad/vigregroup/vigre05/mb.pdf