Stack the vectors up so it's a matrix and take the determinant. The sign tells you which one it is.

No, it does not. The determinant tells you whether two bases have the same or different handedness, not which one is "left" or "right".

It's 2 cosets, one is arbitrarily left handed and the other arbitrarily right handed. If you are in an orientable space :) if not, then there's no global concept of left or right. a

Formulated differently, you cannot determine left- and right-handedness but you can determine same-handedness.

The sentence you quote actually gives you the answer to your question but it is not completely obvious why. :)

The complex numbers are essentially the theory of 2D-space. You are asking about 3D-space. The statement that you quoted tells you that you cannot distinguish between up and down in theory.

Now, 2D-space is part of every 3D-space. There are multiple ways to see this. The easiest is to just drop the z-coordinate.

Suppose you could distinguish left-handed and right-handed in 3D-space. In this case, you would have a way to distinguish up and down in the embedded 2D-space. However, you cannot do this distinction in 2D-space and therefore you cannot distinguish left-handed and right-handed in 3D-space.

No, it's impossible, even in principle, to answer that question. You can draw a picture for either answer.

"Left" and "right" are a dipole. Neither one can exist without the other, and they are symmetric. It's the same issue as we have with the conjugates discussed in Galois Theory.

In fact, in an algebraic (non-ordered/arithmetic/analytic) perspective, it's misleading to use the symbols + and - to label the conjugates in field extensions like sqrt2 and i. Left and Right are better names than + and - for those conjugate pairs. Only when we impose an arithmetic ordering (which is not needed in the theory of algebraic equalities) is it meaningful to use + and -: -sqrt(x) < 0 < +sqrt(x), where x is a positive real number.

(and when x is a negative real number, we immediately see the problem with - again: -i and i are not separable via ordering with respect to 0.)

Once you get to higher extensions where the roots are indistinguishable w.r.t. the base field, mathematicians will just write something like

“Let a_1, …, a_5 be the five roots of p(x).”