For the first question the short answer is that the map $x$ factorises
canonically through the classifying stack $BG$ with $G = Stab(x)$ and the
category of quasi-coherent modules on $BG$ is equivalent to the category
of $G$-vectorspaces over $k$.

Explicitly, you can see the action if you view $F$ as a sheaf on big fppf site on $X$. This site has the category $Sch/X$ of schemes over $X$ as underlying category. The map $x$ is given by a functor $x:Sch/k \to Sch/X$ and $Stab(x)$ is the sheaf of groups defined by $u \mapsto Aut(x(u))$, where
$u:T \to Spec\ k$ is an object in $Sch/k$. Now $F \otimes k = x^{-1} F$ is just the sheaf given by the composition $F \circ x$. For any $u$ as above, we have an action of $Aut(x(u))$ on $F(x(u))$, simply because $F$ is a functor. This gives $F \otimes k$ the structure of a sheaf of $G$-sets. In fact, the construction gives a functor from sheaves of sets on $X$ to sheaves of $G$-sets on $Spec\ k$. We also get a functor to the sheaves of $O_X$-modules to sheaves of $k[G]$-modules which preserves the property of being quasi-coherent.

Note that the above description is very general. For instance, you could take any scheme instead of $Spec\ k$ and any, not necessarily algebraic, stack instead of $X$.

I don't think it is possible to give a general answer to the second question as you stated it. It might be a good idea to have a look on the definition of the push-forward functor. You could start with the push-forward of set-valued sheaves (http://stacks.math.columbia.edu/tag/00XF). A good baby example to consider is to take a group homomorhism $G \to H$ and letting $X = BG$ and $Y = BH$. Verify that pulling back a sheaf from $Y$ to $X$ corresponds to restricting the group action and pushing forward from $X$ to $Y$ corresponds to induction. In particular, if $H$ is trivial, you get the sheaf of invariants. This example also works for quasi-coherent sheaves. Then pull-back and push-forward corresponds to restriction and induction of $G$-representations. In general, the push-forward functor for set-valued sheaves does not coincide with the push-forward functor for quasi-coherent sheaves. In the example with the classifying stacks it works since the map $f\circ x$ is flat.

Surjectivity does not imply surjectivivty on stabilisers. In the example above, the maps are surjective regardless of choice of map $G \to H$ since the underlying topological spaces of classifying stacks are just one point sets. But there are of course group homomorphisms which are not surjective. I'm not sure what you mean by that every action on $f_*F\otimes k$ "comes from" an action on $F$. There is no action on $F$. Just on $F \otimes k$, and it is uniquely determined by $F$.