Deriving Pauli Matrices How does one derive using, say, the operator formula for reflections
$$ R(r) = (I - 2nn^*)(r),$$
the reflection representation of a vector
$$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + yR(\hat{j}) + zR(\hat{k}) 
= xs_x + ys_y + zs_z \\ = x \left[ \begin{array}{ c c } 0 & 1 \\ 1 & 0 \end{array} \right] + y\left[ \begin{array}{ c c } 0 & -i \\ i & 0 \end{array} \right] + z \left[ \begin{array}{ c c } 1 & 0 \\ 0 & - 1 \end{array} \right] = \left[ \begin{array}{ c c } z & x - iy \\ x+iy & - z \end{array} \right] 
$$
that comes up when dealing with spinors in 3-D? Intuitively I can see the matrices are supposed to come from the following geometric picture:

The first Pauli matrix is like a reflection about the "y=x" line. The
  third Pauli matrix is like a reflection about the "x axis". The second
  Pauli matrix is like a 90° counterclockwise rotation and scalar
  multiplication by the imaginary unit
  https://en.wiktionary.org/wiki/Pauli_matrix

but why and how did we make these choices? I know we're doing it to end up using a basis of $su(2)$, but assuming you didn't know anything about $su(2)$, how could you set this up so that it becomes obvious that what we end up calling $su(2)$ is the right way to represent reflections? The usual ways basically postulates them or show they work through isomorphism or say the come from the fact a vector is associated with the matrix I've written above without explaining where that came from. The closest thing to an explanation is that they come from the quaternionic product whose link to all this, especially something as simple as reflections through lines, escapes me. 
 A: Lucubration needs not light with insight. I fear you are expecting to make lemonade with apples. Here is why. 
The basic relation is the multiplication law of two Pauli vectors predicated on the abstract properties of the Pauli matrices, not their particular realization, 
$$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) \, I + i ( \vec{a} \times \vec{b} )\cdot \vec{\sigma} ~.$$
Any similarity transformation by an invertible matrix $\vec{\sigma} \mapsto U^{-1} \vec{\sigma} U$ preserving this relation would likewise lead to the fundamental reflection formula you miswrote,
$$
- \hat{n}\cdot \vec{\sigma} ~~ \vec{x} \cdot \vec{\sigma} ~~\hat{n}\cdot \vec{\sigma} = ((1\!\!1 -2 \hat{n}\hat{n}) \cdot \vec{x}) \cdot \vec{\sigma}. 
$$
The missing 2 is crucial in your first formula, if it is to yield a reflection and not a projection. (The first R you wrote is a 3x3 matrix acting on a vector. The second one is a 2x2 matrix representing a different Pauli vector.) This is the Householder formula, of course, acting on a 3-vector properly mapped to the suitable Pauli vector, a 2x2 matrix. (It is used routinely in the adjoint rotation of a Pauli vector, thus seen to amount to the double-angle rotation of the vector rep.) 
And that's that. Edit: since you asked for an illustration, taking the axis perpendicular to the plane of reflexion (x,z) to be along y, so n=(0,1,0), the above formula reduces to just 
$$
-\sigma_y ~~\vec{x} \cdot \vec{\sigma} ~~\sigma_y= x \sigma_x -y \sigma_y +z\sigma_z, 
$$
the celebrated conjugation of the pseudoreal rep of SU(2). 
The above generic 3d reflections have nothing to do with the somewhat fanciful mnemonic rules of the wiktionary you quote for the representation-dependent three Pauli matrices in terms of 2d, plane reflections across lines, not planes, as any similarity transform of the standard Paulis would do, as seen above! The (somewhat frivolous) mnemonic summarizes, "for desert island use", plane transformations that reflect a 2-vector according to the standard convention Pauli matrices. It is no more than a convention though---admittedly the easiest one. You could permute them cyclically and so get the cyclically permuted mnemonic. Since this is aggressively representation-dependent, it amounts to a case where one could safely reassure oneself that there is no deep geometrical connection involved---a rare occasion, indeed. 
A: A rotation is of the form $$\begin{bmatrix} \cos(\theta) & - \sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$$ A reflection is of the form $$\begin{bmatrix} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & - \cos(2\theta) \end{bmatrix}$$
If we want to find a $2-$dimensional representation of a $3-$dimensional rotation then we can assume $$\mathbb{R}^3 = \mathbb{R} \times \mathbb{C}$$
so that $$(x,y,z) \mapsto (z,x+iy)$$
Thus in a space where all vectors are of the form $(z,x+iy)$, the column vectors of a rotation matrix will be of this form, with the second column orthogonal to the first, thus a reflection matrix would be $$\begin{bmatrix} z & x-iy \\ x+iy &  - z \end{bmatrix}$$
Mapping $(1,0,0) \mapsto (0,1)$ etc... gives the Pauli matrices. 
