Why does the dipole moment depend on the distance? Why does the dipole moment of an electric dipole of different charges depend on the distance from the origin? Physically, I don't understand why something that measures how much of a dipole something is should depend on how far away that thing is from the origin.
 A: The electric dipole moment is defined as
$$p = \int r \;\mathrm dq$$ 
In the case of a pair of charges for which both charges are of the same magnitude, the choice of the origin turns out to be irrelevant:
$$ p = \mathbf{r_1} q - \mathbf{r_2} q = q(\mathbf{r_1} - \mathbf{r_2}) = q\mathbf{d}$$
where $\mathbf{d}$ is the distance between the charges. However, when the two charges are not of equal magnitudes, $q$ cannot be factored out of the expression, so the choice of origin is of significance. In general, it is the case (as you noted) that dipole moment depends on a chosen point of reference.
Physically speaking, the dipole moment - like moment of inertia - is a property of a system that's dependent on the reference point you chose. This is intuitively true if you consider the following:
$$\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$$
To calculate torque requires a specified axis of rotation. For the case of a pair of charges with equal magnitude, the torque is independent of the chosen axis of rotation. However, this isn't the case for non-neutral dipole, so for $\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$ to hold, $\textbf{p}$ must also be dependent on the point of reference chosen.
A: If the total charge of the system is zero, the dipole moment does not depend on distance.
much same like:
if total momentum of a system is zero, the angular momentum does not depend on the origin of reference.
Dipole moment is the intrinsic property of a system (subtract total charge to zero first);
Angular momentum is the intrinsic property of a system (the reference is the system mass center);
