Yeah, by "minimal" I mean in the to say that we impose the relation $K<L$ if there exists a map $K\to L$. (In which case $k(x) \cong k(x^2)$.) But you're absolutely right about there being more extensions with finite automorphism group. Instead, I think you can characterize finite extensions by the property that they have finitely many sub-extensions.

I'm not sure what you mean by your part 1. I think you can say something along these lines, but the statement that there is a map $K\to L$ if $L$ has higher transcendence degree is in general false. (There are no maps from an extension of $\mathbb{F}_p$ to the function field of $\mathbb{P}^1_{\mathbb{F}_p}.$)

note that this hypothesis is essential: a morphism ramified at 0, 1, and $\infty$ is determined by combinatorial data, and can be shown to be defined over $\overline{\mathbb{Q}}$

"isometries of R^n" isn't very clear. If you want isometric maps from the hypercube to the sphere, there are none because a sphere has nonzero curvature and the hypercube is flat.

With a little more background, this question can be asked in an appropriate stackoverflow. This is not a research-level mathematics question.

That is a very good point. @WeatherReport, if you were interested in $f(x,y,t) + f(x^{-1}, y, t^{-1})$ then the above derivation should work (with $x$ replaced by $y$). Are you sure that what you want is $f(x,y,t) + f(x^{-1}, y, t)$?

Give me an area of mathematics that doesn't care about K theory :)