How to understand group action especially Galois action on a scheme? I read a lot of books and find none of them give explicit descriptions of group action on schemes. I am very confused now and have lots of questions. So I think these questions will be relatively long and hope you could read them.
First, in many books, these is such a sentence: let $X=\mathrm{Spec}(A)$, then a finite group $G$ acts on $X$ on the right is  equivalent to $G$ acts on $A$ on the left.
I try to write this explicitly. $\Longrightarrow$ Suppose $G$ acts right on $X$, then we have group homomorphism: $\phi: G^{op}\to \mathrm{Aut}(X)=\mathrm{Aut}(\mathrm{Spec}(A)), g\to \phi_{g} $. $\phi_{g}$ induces a ring homomorphism $\hat{\phi}_{g}\in \mathrm{Aut}(A)$. Then we could get a group homomorrphism $\psi: G\to \mathrm{Aut}(A)$. Thus we have right action $G$ on $A$.$\Longleftarrow:$ similarly.  Is this understanding right?
Secondly, when we say a group acts on a scheme, do we mean left action or right action? I feel often we mean left action, that is a group homomorrphism $G\to \mathrm{Aut}(X)$. Am I right?
Thirdly, how should I understand Galois action on an affine variety? For example, let $G=Gal(\mathbb{Q(i)/\mathbb{Q}})=\{1, \sigma\}$ and $H=Gal(\bar{\mathbb{Q}}/\mathbb{Q})$, $\tau$ is a non-trivial element of $H$.  $X=\mathrm{Spec}(\mathbb{Q}[x])$. How to describe $\sigma,\tau$ acting on $X$? Suppose $\sigma,\tau$  corresponds $\phi_{\sigma},\phi_{\tau}\in \mathrm{Aut}(\mathbb{Q}[x])$, then what are $\phi_{\sigma},\phi_{\tau}$? It seems that rational points do not move, but some other points move. I do not know how to describe it.
Fourth, how to understand Galois action on a non-affine variety? For example, using the same notion with the third, but let $X=\mathbb{P}_{\mathbb{Q}}^{1}$ or $\mathrm{Proj}(\mathbb{Q}[x,y,z]/(y^2z-x^3-xz^2))$, how to describe the action $\sigma,\tau$ on $X$ exactly? For general projective variety $X$, should we first embed it to some projective space and them describe Galois action on it?
Lastly, how to understand the Galois action on morphism of varieties? That is let $X$ and $Y$ be two varieties over $k$,  $\phi: X\to Y$ is a morphism between them. Let $G=Gal(\bar{k}/k)$, $\sigma$ is an element of $G$, then what is the action of $\sigma$ on $\phi$, $\phi^{\sigma}$? Many books just write it and never explain it. For example ,let $X=Y=\mathrm{Spec}(\mathbb{Q}[x])$ and $\phi$ corresponds the ring map $x\to x^2$, then what is $\phi^{\sigma}$? If $X=Y=\mathrm{Spec}(\mathbb{Q}(i)[x])$, $\phi$ also corresponds $x\to x^{2}$, , then what is $\phi^{\sigma}$?
Thank you very much for reading these questions. Could you explain these to me? Also, if you have some good examples to help understanding group action, please write them down.
 A: 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.
