Serre found examples of surfaces defined over number fields whose homotopy type depends on the embedding of the field into the complex numbers. This might answer in the negative your question, after replacing the rational numbers by a number field.

This question appears close to weak approximation on the moduli space of curves. Thus, while it may be true for low genus (e.g. if the moduli space of curves is (uni-)rational),it is hard to imagine that this is true for large genus (e.g. when the moduli space is of general type).

Indeed, as @LouisDeaett says, I only meant it as a one-sided test. Still, you can squeeze out a little bit more, if you do have solutions satisfying the few minors you tested. Such solutions correspond to matrices with some minors forced to be zero. If you are lucky (which has often been the case in my experience), then you may find that the matrices actually have small rank.

Rather than using all minors of a fixed size, often choosing a few randomly chosen ones is enough. If you get the empty scheme, then you are done. Otherwise, you might be able to find points which have a chance of having small rank...

You can try to recover the gonality of the curve by looking at how the number of points behaves over extensions of $K$ of a fixed degree. For instance, lots of quadratic points should imply that the curve is hyperelliptic.

A quick experiment suggests that the map is bijective if and only if $n$ is a power of $2$.

en.wikipedia.org/wiki/List_of_Greek_phrases

For Q2, have you tried $n$ even and $R=\mathbb{F}_2$?