There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in the field.
Question 1: Let $(A, A^+)$ be a complete sheafy Tate-Huber pair and suppose that $X=\operatorname{Spa}(A, A^+)$ is covered by some open affinoids $X_i=\operatorname{Spa}(A_i, A_i^+)$ for some strongly noetherian Tate-Huber pairs $(A_i, A_i^+)$. Does it imply that $A$ is strongly noetherian as well?
Question 2: Let $(A, A^+)$ be a complete strongly noetherian Tate-Huber pair, and let $X=\operatorname{Spa}(A, A^+)$ be its adic spectrum. Is it true that local rings $\mathcal O_{X,x}$ are noetherian for all $x\in X$?
Remark: I think that this is known to be true in the case $(A, A^+)$ is topologically finite type over a complete affinoid field $(k, k^+)$. The case of rk-$1$ affinoid fields is explained here. Basically the same proof generalizes to all complete affinoid fields.
Before asking question $3$ I need to introduce some terminology (that might be non-standard). Let us say that a morphism of adic spaces $f:X \to Y$ is ${\it flat}$ if $\mathcal O_{y, f(x)} \to \mathcal O_{X, x}$ is flat for all $x\in X$.
Question 3: Let $f:(A, A^+) \to (B, B^+)$ be a continuous morphism of strongly noetherian Tate-Huber pairs, such that $f:A \to B$ is flat. Does it imply that $f^*:\operatorname{Spa}(B, B^+) \to \operatorname{Spa}(A, A^+)$ is flat?
(UPD) Remark 2: It suffices to check this question only at rk-$1$ point $x\in \operatorname{Spa}(A, A^+)$ due to Lemma 1.1.10(iii) from Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces". Part (iv) of the same Lemma implies that $f(x)$ is also a rk-$1$ point of $\operatorname{Spa}(B, B^+)$.
Question 4: Let $f:(A, A^+) \to (B, B^+)$ be a continuous morphism of strongly noetherian Tate-Huber pairs such that $f:A \to B$ is flat. Is it true that the induced map of ``rational domains'' $$ A\langle \frac{f_1}{g}, \dots, \frac{f_n}{g} \rangle \to B\langle \frac{f_1}{g}, \dots, \frac{f_n}{g} \rangle $$ is also flat for any elements $f_1, \dots, f_n, g\in A$ generating the unit ideal?
Remark 3: I believe that I checked that the positive answer to Question $4$ implies the positive answer to Question $3$. But it is not clear how to prove it. The issue is that $$ B\langle \frac{f_1}{g}, \dots, \frac{f_n}{g} \rangle = A\langle \frac{f_1}{g}, \dots, \frac{f_n}{g} \rangle \widehat{\otimes}_A B, $$ and not just $$ A\langle \frac{f_1}{g}, \dots, \frac{f_n}{g} \rangle \otimes_A B. $$
