Let not worry about the char p (suppose it is much larger than the automorphism order). As, in our case, we are dealing with the group generated by one automorphism, hence is always cyclic. Now how repn theory can help me to find the generators? (I don't think repns are too helpful when our group is cyclic)

Part 1 and 2 of the question were actually to derive x^q -x and (x^q - x)^(q-1), resp.

What if the function field is not rational? I see that if the (deg(x), deg(y)) = 1 then I can generalize it to non-rational case. But the question, now is that how to find such generators (without computing the RR spaces, because that seems to me too much for such a small thing)

@Will-sawin, Me neither. I was desperate so I just tried different silly stuff, so hopefully something illuminating might happen. Later, I found out the reason of that phenomena: it's because [F:F2] = 2 and hence F is hyperelliptic, so we have that all rational sub field of F up to some degree are all in F2. then deg(v2/v1) <= |(v1)| 2g-2. and hence v2/v1 in F2. But still my question is that if there's anything special about those rational fields or we are just generating random rational function fields, in that case, the question is that what's the attraction of Felipe algorithm then?

@Felipe, "something that will factor through the Jacobian of $\mathbb{P}^1$". The answer is easy in this case because, we know all zero degree divisors over $\mathbb{P}^1$ are principal. I just brought it as an example, for testing a general algorithm. I'll replace it with a less toyish example.

@Felipe, the function field to be computed is the function field of $\mu \circ \phi(C)$ which is an image of a curve, hence is a curve and has function field of transcendental degree 1. My guess was that $Jac(\mu \circ \phi(C)$ is $\mu(Jac(C))$ and hence $K^\mu$ would the function field I mentioned in the comment for $\mu(Jac(C))$. I changed the title to be more appropriately represent my question.