To answer your second question first: an orthogonal matrix $O$ satisfies $O^TO=I$, so $\det(O^TO)=(\det O)^2=1$, and hence $\det O = \pm 1$. The determinant of a matrix tells you by what factor the (signed) volume of a parallelipiped is multipled when you apply the matrix to its edges; therefore hitting a volume in $\mathbb{R}^n$ with an orthogonal matrix either leaves the volume unchanged (so it is a rotation) or multiplies it by $-1$ (so it is a reflection).
To answer your first question: the action of a matrix $A$ can be neatly expressed via its singular value decomposition, $A=U\Lambda V^T$, where $U$, $V$ are orthogonal matrices and $\Lambda$ is a matrix with non-negative values along the diagonal (nb. this makes sense even if $A$ is not square!) The values on the diagonal of $\Lambda$ are called the singular values of $A$, and if $A$ is square and symmetric they will be the absolute values of the eigenvalues.
The way to think about this is that the action of $A$ is first to rotate/reflect to a new basis, then scale along the directions of your new (intermediate) basis, before a final rotation/reflection.
With this in mind, notice that $A^T=V\Lambda^T U^T$, so the action of $A^T$ is to perform the inverse of the final rotation, then scale the new shape along the canonical unit directions, and then apply the inverse of the original rotation.
Furthermore, when $A$ is symmetric, $A=A^T\implies V\Lambda^T U^T = U\Lambda V^T \implies U = V $, therefore the action of a symmetric matrix can be regarded as a rotation to a new basis, then scaling in this new basis, and finally rotating back to the first basis.
The term "dual complex numbers" is misleading.
Let $\mathbb D$ denote the dual numbers. Then the dual quaternions are defined as $\mathbb D \otimes \mathbb H$. In other words, they are exactly the quaternions with dual number components instead of real components. The "dual complex numbers" correspond to the subalgebra spanned by $$\{1,i,\epsilon j, \epsilon k\}$$
[EDIT: The following account of the dual complex numbers is based on them being a subalgebra of the dual quaternions. I make use of the idea that the dual quaternions are effectively an "infinitesimal thickening" of the quaternions. We can use this to study what happens if we vary the axis and angle of a rotation by an infinitesimal amount. Understanding this requires some familiarity with the quaternions, and some comfort with the "infinitesimal" interpretation of the dual numbers. I do not make use of the fact that the dual quaternions can be used to express rigid body motions in 3D.]
Consider the "infinitesimal plane" $$\Pi = \{i + x\epsilon j + y\epsilon k: x \in \mathbb R, y\in \mathbb R\}$$
Rotations about $i$, represented by the quaternions $e^{i\theta/2}$, rotate this plane back onto itself. This is how to represent the rotations about the origin of a plane using dual-complex numbers.
Rotations about an axis $A j + B k$ (where $A, B \in \mathbb R$) which is perpendicular to $\Pi$, by the infinitesimal angle $\epsilon$, are represented by the dual complex numbers $e^{\frac 1 2 (A \,\epsilon j + B\, \epsilon k)}$. These also map $\Pi$ back onto itself. Points on $\Pi$ experience a translation.
Those are two extreme cases.
In general, each dual complex number has one of two polar forms:
$e^{(i+ A\, \epsilon j + B\, \epsilon k)\theta/2} = \cos(\theta/2) + \sin(\theta/2)(i+A\, \epsilon j + B\, \epsilon k)$. From a planar point of view, these represent a rotation of angle $\theta$ about the point $(A, B)$. From a 3D point of view, $i+A\, \epsilon j + B\, \epsilon k$ is the axis of rotation, and $\theta$ is the angle.
$e^{\frac 1 2 (A \,\epsilon j + B\, \epsilon k)}=1+\frac 1 2(A\, \epsilon j + B\, \epsilon k)$. These represent the translations. The 3D and 2D points of view are described above.
In a sense, a translation is equal to a rotation about a point at infinity. To illustrate this, consider the following limit of the first polar form: $$\lim_{S \to \infty} e^{(i+S(A\, \epsilon k + B \epsilon k))\theta/(2S)} = e^{(A\, \epsilon j + B\, \epsilon k))\theta/2}$$ The result has the second polar form. Note that in order to obtain the desired limit, the angle of rotation has to get shrunk at the same rate as the centre of rotation goes to infinity, otherwise the arc of rotation grows to an infinite length.
Best Answer
Dual numbers, unlike complex numbers, illustrate Galileo's invariance principle. The infinitesimal part of a dual number represents the velocity.
In particular, whereas the multiplication of two complex numbers can be understood as a combination of scaling and rotation, the multiplication of dual numbers is actually equivalent to a scaling and shear mapping of plane, since $(1 + p \varepsilon)(1 + q \varepsilon) = 1 + (p+q) \varepsilon$. You can see that the classical velocity addition law emerges.
The "hyperbolic" multiplication law of special relativity (namely the corresponding velocity addition law $v\oplus u=(v+u)/(1+vu)$) requires Lorentz transformations, which can be packaged into different types of numbers like, for example, quaternions (see Generalized complex numbers).