Let $F$ be an $\mathcal{O}_X$ module. Given a global section $s\in F(X)$, one can define $\alpha: \mathcal{O}_X\to F$ as follows:
\begin{equation}
\alpha(U): g\mapsto g.s|_U
\end{equation}
Here I mean by $g.$ the action of $\mathcal{O}_X$ on $F$.
Conversely given a homomorphism $\alpha: \mathcal{O}_X\to F$, define
\begin{equation}
s=\alpha(X)(1)\in F(X)
\end{equation}
where $1\in \mathcal{O}_X(X)$.
You can check that these two constructions are inverse to each other.
Note that $X$ does not need to be a projective space.
Your attempt at writing things down more explicitly is good. The important thing to realize is that because the equivalence between affine schemes and rings is contravariant, the order of composition changes and swaps left/right actions.
If the authors do not specify whether an action is a left/right action, there is a good chance it is either inferrable from the context, or does not matter. If you can point to specific examples where you think it does matter and it's not clear, these would be good things to ask about as a separate question.
For your explicit example involving actions on $\operatorname{Spec} \Bbb Q[x]$, note that all automorphisms you mention fix $\Bbb Q[x]$ (because they fix $\Bbb Q$) and so they're the identity map. If you instead talk about $\operatorname{Spec} \Bbb Q(i)[x]$, then you do see some movement happening: $(x-i)$ is swapped with $(x+i)$, for instance. One classic fact to learn (mentioned early on in Vakil, for instance) is that if $k\subset K$ is a Galois extension, then $Gal(K/k)$ acts on $\Bbb A^n_K$ and the orbits are precisely the points of $\Bbb A^n_k$. Another good thing to know is that if we have an automorphism $\sigma:k\to k$, then the induced action of $\sigma$ on the $k$-rational points of $\Bbb A^n_k$ is $(a_1,\cdots,a_n)\mapsto (\sigma(a_1),\cdots,\sigma(a_n))$ (proof: just write down the action on the maximal ideal $(x_1-a_1,\cdots,x_n-a_n)$). So these two facts should give you a full understanding of what the Galois action does on affine space.
For subvarieties of affine space, you may need to be a little careful when you define actions. If you want to define an action on $V(I)\subset \Bbb A^n_k$, then you'll need the Galois action to fix $I$. For an example of why this is necessary, think about $V(x-i)\subset \operatorname{Spec} \Bbb Q(i)[x]$. The Galois action here doesn't fix this subvariety and thus does not define an automorphism of it.
As far as group actions on non-affine schemes, things can get a little hairy depending on the specific context. Fortunately, in the case of the Galois action on a scheme $X$ over a field $k$, everything is induced from what happens on $\operatorname{Spec} k$: we define the action of $\sigma \in Gal$ on $X$ to be the map $X\times_k \operatorname{Spec} k\cong X\to X$ induced by taking the fiber product of $X\to \operatorname{Spec} k$ with the automorphism $\sigma:\operatorname{Spec} k\to\operatorname{Spec} k$. If you're interested in studying this action on $X$ via an embedding $X\to Y$, then you need to make sure that the embedding respects the action (this is often described in the literature as "the morphism is an intertwiner" or the like - it means that the morphism commutes with the automorphism).
This brings us to your final point - when we have a morphism of varieties and we want to think about group actions on the source and target, it's usually important to us that the morphism respects the actions: if $f:X\to Y$ is our morphism, we want $g\cdot f(x)=f(g\cdot x)$. Sometimes we don't mean this, though, and if we have an action on $X$, then we can get an action on $\operatorname{Hom}(X,Y)$ by precomposing a map $f:X\to Y$ with an automorphism $\sigma:X\to X$, or if we have an action on $Y$, then we can get an action on $\operatorname{Hom}(X,Y)$ by postcomposing a map $f:X\to Y$ with an automorphism $\sigma:Y\to Y$. So the "action on a morphism" you mention in the comments is just a version of this. (Specifically, saying that $\sigma(F)=F$ is the second version of this - $\sigma$ acts on $Y$, which induces an action on maps as described, if you want to think of it this way. I usually don't, though this isn't my main area and I'm a mathematician, not a cop.)
As for the descent question, the goal is to have $\phi_{\sigma\tau}:(\sigma\tau)V\to V$ factor as $\sigma(\tau V)\stackrel{\sigma\circ\phi_\tau}{\longrightarrow}\sigma V\stackrel{\phi_\sigma}{\longrightarrow} V$, which is just the cocycle condition from (non-Galois) descent.
Best Answer
To answer your direct question, it is useful to first abstract the situation slightly. That said, maybe the following will be helpful to help orient you to the rest of the answer.
Preamble
Since you seem to be confident with Grothendieck topologies, let me assume that you know what I mean by a site.
(NB: All of this is actually more cleanly phrased at the topos level, but that level of abstraction isn't perhaps helpful to address your specific question.)
I cannot go into too much detail in this post due to time and space constraints. Perhaps these notes of mine could be helpful if you want a more intuitive explanation (I apologize they are quite old, and perhaps not quite succinct or maximally insightful).
If you are interested in trying to pursue the material here more seriously, let me recommend some texts.
I would suggest either [Bertin] (most relevantly here being [Bertin, §1]) and [Olsson] (most specifically [Olsson, Chapter 2 -- Chapter 4]).
That said, except for maybe getting comfortable with the idea of a stack (which I can't expound on here), I hope the post is self-contained.
Torsors
Let us first be precise about what we mean by 'torsor' here. This is particularly relevant, as people sometimes (in my opinion confusingly) confuse the terms 'principal homogenous space' (i.e., a representable torsor) and 'torsor'.
So, let us fix the following data:
If $\mathcal{F}\colon \mathbf{C}^\mathrm{op}\to\mathbf{Set}$ is a sheaf of sets on $\mathbf{C}$, then by a right action of $\mathcal{G}$ on $\mathcal{F}$ I simply mean a morphism of sheaves of sets $\mathcal{F}\times\mathcal{G}\to \mathcal{F}$ such that for all objects $X$ of $\mathbf{C}$ the induced map $\mathcal{F}(X)\times\mathcal{G}(X)\to\mathcal{F}(X)$ defines a right action of the group $\mathcal{G}(X)$ on the set $\mathcal{F}(X)$.
(NB: You can of course axiomatize this at the sheaf level (without referring to the values to varying points) by using Yoneda's lemma, but I find that harder to remember precisely.)
To define torsors, it's useful to develop three extra pieces of terminology:
We then have the following easy proposition, which is left to you as an exercise.
(NB: As per usual, the way that one might think of a $\mathcal{G}$-torsor is as a 'space' with a $\mathcal{G}$-action which is 'locally trivial'.)
A morphism of $\mathcal{G}$-torsors is just a $\mathcal{G}$-equivariant morphism of sheaves (which is automatically an isomorphism). We then make the following notational definitions.
Then, as you have already implicitly pointed out one can understand the pointed sets $\mathrm{Tors}(\mathcal{G})$ and $\mathrm{Tors}_\mathcal{U}(\mathcal{G})$ (with neutral element given by the isomorphism class of the trivial $\mathcal{G}$-torsor) cohomologically.
All of this was likely already known to you, but setting our notation and terminology will make things smoother below.
Stacks
As is implicit in your question, often in nature sheaves of groups are presented to us as automorphism sheaves of some mathematical object. To make the notion of an 'automorphism sheaf' precise, and to present the generality in which one can make strong connections between 'twists' and torsors for automorphism sheaves, we need to introduce the notion of stacks.
Prestacks
To begin, let us recall that a groupoid is a category where every morphism is invertible. We denote the category of groupoids by $\mathbf{Grpd}$, where morphisms are just natural transformations.
So, in essence to every object $X$ of $\mathbf{C}$ we have a groupoid $\mathscr{M}(X)$, and for every morphism $Y\to X$ we have a natural transformation $\mathscr{M}(X)\to \mathscr{M}(Y)$.
In practice, such prestacks arrive by considering
Let us give some classic examples of prestacks in algebraic geometry. In what follows, for a scheme $S$ and $\tau\in\{\mathrm{fppf},\mathrm{\acute{e}t},\mathrm{Zariski}\}$ let $S_\tau$ denote the small $\tau$-site over $S$.
There are endless examples, but I leave it to you to consult textbooks for more.
Stacks
Now, one can think of a prestack as a 'higher category' (specifically $2$-category) version of a presheaf. There then is, as expected, the analogue of a separated presheaf/sheaf.
Some words are in order. First of all, this looks a priori quite different from the usual definition of sheaves. But, this is deceptive as the following instructive exercise shows.
In other words, this exercise shows that if $\mathcal{F}$ is just a normal presheaf as opposed to a prestack (i.e., valued in sets as opposed to groupoids) then the triple product doesn't matter, and we essentially recover the normal condition for a presheaf to be a sheaf.
So why do we need to consider triple products in the case of prestacks, and what is this funny 2-limit (as opposed to normal limit)? Both of these have to do with subtleties which occur when considering groupoids and not sets.
To start, this 2-limit is a 'stronger' notion of limit, where we not only require that the objects on the $X_i$ are isomorphic when restricted to each double product (note that 'equality' is essentially a useless notion in a non-discrete groupoid!), but that we remember this isomorphism, and the triple product is needed to ensure that these isomorphisms agree on triple overlaps.
In more concrete terms, one can think of this 2-limit as the category of $\mathscr{M}$-descent data for the cover $\mathcal{U}$ (see [Bertin, §1.2.4]) -- thus the notation $\mathbf{DD}_\mathbf{C}(\mathcal{U})$.
Let us now discuss whether the previous examples of prestacks are stacks.
Example 12 ([Olsson, Theorem 13.1.2]): The prestack $\mathscr{Ell}$ is always a stack, with no restrictions on $S$ or $\tau$.
Example 13 (see Tag 0CCH and Tag 08KE): The prestack $\mathscr{QP}$ is a stack if $S$ is the spectrum of a field, and $\tau\in\{\mathrm{\acute{e}t},\mathrm{Zariski}\}$ but often fails to be a stack in other situations (except when $\tau$ is the Zariski topology).
Example 14 (see Tag 0247): The prestack $\mathscr{QA}$ is a stack.
Example 15 (see Tag 0241): The prestack $\mathscr{S}$ is a separated prestack, but is rarely a stack.
(NB: For instance, the 'stackification' of $\mathscr{S}$ for the étale topology is (essentially) the category of algebraic spaces.)
Example 16 (see [Bertin, Proposition 2.17 and Exercise 2.18]): The prestack $\mathrm{Bun}_n$ is a stack.
Example 17 (see [Giraud, Chapitre III, §1.4.4.]): The prestack $\mathbf{Tors}_\mathcal{G}$ is a stack.
Twists
To relate stacks and torsors, we begin with a useful recharacterization of when a prestack is separated in terms of so-called Isom-presheaves.
Namely, suppose that $\mathscr{M}$ is a prestack over $\mathbf{C}$ and $A$ and $B$ are objects of the groupoid $\mathcal{M}(X)$, where $X$ is an object of $\mathbf{C}$. We then have a presheaf $$\mathrm{Isom}(A,B)\colon\mathbf{C}_{/X}^\mathrm{op}\to \mathbf{Set},$$ which associates to any $Y\to X$ the set of isomorphisms $A|_Y\to B|_Y$ in $\mathscr{M}(Y)$.
In particular, for a separated prestack $\mathscr{M}$ and an object $A$ of $\mathscr{M}(X)$ we get a group sheaf on $\mathbf{C}_{/X}$ given by $\mathrm{Aut}(A):=\mathrm{Isom}(A,A)$.
We denote the isomorphism classes (in $\mathscr{M}(X)$) of twists of $A$ by $\mathrm{Tw}(A)$ and $\mathrm{Tw}_\mathcal{U}(A)$ respectively. These are naturally pointed sets, with the neutral element being the trivial twist $A$ itself.
It is not hard to see that there are natural maps of pointed sets
$$\mathrm{Tw}(A)\to\mathrm{Tors}(\mathrm{Aut}(A)),\qquad \mathrm{Tw}_\mathcal{U}(A)\to \mathrm{Tors}_\mathcal{U}(\mathrm{Aut}(A))\qquad (\ast),$$
given by sending $B$ to $\mathrm{Isom}(A,B)$ with the natural right-action by $\mathrm{Aut}(A)$. Of course, there is no reason a priori that these are bijections. But, we have the following beautiful proposition.
So, as a corollary of this, we obtain the following.
Conclusion
To answer your original question, we see that we have given a sufficient condition for your map
$$\phi_\mathcal{U}\colon \mathrm{Tw}_\mathcal{U}(\pi\colon Y\to X)\to \check{H}^1(\mathcal{U},\mathrm{Aut}(A)),$$
to be a bijection: that $\pi$ belongs to some presubstack $\mathscr{M}\subseteq \mathscr{S}$ (with $\mathscr{S}$ as in Example 7) such that $\mathscr{M}$ is a stack on $S_\tau$.
While this doesn't provide an overly satisfying answer, it very much allows you to get a sense of what can go wrong, and gives you a way of trying to find references for the result you are looking for.
For instance, the reason that you see twists of quasi-projective varieties on the small étale site of $S$, where $S$ is the spectrum of a field, is that there the theory works well (it often goes under the heading of 'Galois descent for quasi-projective varieties)). The fact that such quasi-projective morphisms rarely form a stack in greater generality is one of the reasons you were having difficulty finding literature on it -- the theory doesn't work well there.
One last thing I'll point out is that the Corollary says that twists are (often) special cases of torsors. But, in fact, the other direction is true as well. Indeed, essentially by definition, torsors are twists of the trivial torsor in the stack $\mathbf{Tors}_\mathcal{G}$. Thus, up to technical points, I think of the theory of torsors and twists as being essentially equivalent. This is an incredibly powerful tool for thinking about both concepts.
References:
[Bertin] Bertin, J., 2013. Algebraic stacks with a view toward moduli stacks of covers. In Arithmetic and Geometry Around Galois Theory (pp. 1-148). Springer Basel.
[Giraud] Giraud, J., 2020. Cohomologie non abélienne (Vol. 179). Springer Nature.
[Olsson] Olsson, M., 2016. Algebraic spaces and stacks (Vol. 62). American Mathematical Soc..