Exponential cosine-screened Coulomb potential I know from various sources that the expression for the exponential cosine-screened Coulomb potential (ECSCP) with screening parameter $~µ~$ is 
$$V(~\vec r~)=\dfrac 1r~e^{-\mu r}~\cos(\mu r)\qquad\text{(in a.u.)}$$
Now I want to know 
$\bf 1.~~$Where did this expression come from (Please provide the mathematical derivation) ? 
$\bf 2.~~$For which purpose it is used for?
$\bf 3.~~$Is there any terminology exists named  exponential sine-screened Coulomb potential ? If yes, indicate its application. 
Here are some sources from where I got the terminology
Bound Eigenstates of the Exponential Cosine Screened Coulomb Potential
Analytical Treatment of the Oscillating Yukawa Potential
 A: I will try to explain where this potential comes from. It appears basically when considering the screened potential created by a fixed ion, in a medium where electrons (fermionic charge carriers) can move. For simplicity, I will just detail what happens in the classical picture, which just gives the "exponential" part of the potential (also known as "Debye" screening).
In natural (Gaussian) units, the potential $V(r)$ created by a static charge distribution $ \rho(r)$ follows the Poisson equation:
$$\Delta V(r) = - 4 \pi \rho(r)$$
For a single point charge in vacuum, $ \rho(r) = \delta(r)$ with $\delta$ the Dirac distribution. This leads to the famous $1/r$ Coulomb potential.
However, in a medium with free (negative) charge carriers, the charge distribution will be modified by the Coulomb potential. Negative charges will accumulate around the positive central charge, screening the potential at long range.
In order to solve the problem, we need a second equation, describing how the local charge density responds to a change in potential. I will separate artificially positive and negative charges, as they respond differently to an electric potential. The simplest possible dependency being linear, I will take $ \rho_-(r) = \rho_0 + \frac{\mu^2}{4 \pi} V(r)$ (density of negative charges, increasing with $V$)$^{\dagger}$, and $ \rho_+(r) = \rho_0$ (density of positive charges, for instance other ions, supposed to be constant).
Reinjecting into the equation, we find:
$$\Delta V(r) = - 4 \pi \delta(r)- 4 \pi (\rho_+(r) - \rho_-(r)) = - 4 \pi \delta(r) +   \mu^2 V(r)$$ 
As the $\delta(r) = 0$ for $r >0$, I will focus on the solution of $V(r)$ for $r>0$. The point charge will only be useful to determinate the constant in front of $V(r)$ at the end. Using the expression of $\Delta V(r) = \frac{1}{r^2} \frac{\partial (r^2 V'(r))}{\partial r}$ for a spherically symmetric potential, and setting $u(r) = rV(r)$, we find (after substitution):
$$\Delta V(r) = \frac{u''(r)}{r} =  \mu^2 \frac{u(r)}{r}$$
Thus $u(r) = A \exp(-\mu r)$ and $V(r) = \frac{A \exp(-\mu r)}{r}$. $A$ can be found using Gauss theorem, but you will not be surprised that the potential is completely dominated by the point charge for $r \to 0$ (if the charge distribution $\rho_-(r)$ is regular), so we must take $A=1$ to find the Coulomb potential for $r\to 0$. In the end, $V(r) = \frac{\exp(-\mu r)}{r}$. This is sometimes called the Debye potential, which can be used for instance to describe the screening of a point charge inside a plasma.
Now what about the $\cos$ part? I believe this arises purely from the fermionic nature of the charge carriers, so it is essentially a quantum phenomenon (I might be wrong). Indeed, because of Pauli's exclusion principle, fermions can not occupate the same momentum state. The effect on the density is a bit subtle, because you have to go from momentum to real space, but this results in so-called Friedel oscillations of the correlations. Basically, if you have a single particle at $r=0$, the probability to find another particle extremely close will be small (due to Pauli's exclusion principle), but after some distance, the probability to find another particle will be increased compared to the homogeneous case.
I am not super familiar with the way to compute the screened potential for fermionic particles, but this results somehow in an extra oscillating term in $V(r)$, such that:
$$V(r) = \frac{\exp(-\mu r)}{r} \cos(k_F r).$$
(Note that in general, $\mu$ and $k_F$ can be different)
In short, the exponential part can be explained mostly by "classical" screening, whereas the oscillating term is related to the fermionic nature of the charge carriers. It can be used for instance to describe a ionic impurity in a cristal.
Regarding your third question, I do not believe that an exponential sine-screened potential would bear much physical signification. One of the feature of the cosine screened potential is that it is equivalent to the Coulomb potential for $r \to 0$. This is because no amount of electrons will be enough to screen completely the Coulomb potential at short distance (because of the $1/r$ divergence).

$^{\dagger}$One of the situations where this might apply is if you assume a "thermal distribution" for the density of electrons $\rho_-(r) = \rho_0 \exp(-E(r)/kT)$, where $E(r) = -e V(r)$ is the local potential energy. For large temperatures, this can be expanded to first order as $\rho_-(r) = \rho_0 + \frac{\rho_0 e}{kT} V(r)$, thus yielding $\mu^2 = 4 \pi \frac{\rho_0 e}{kT}$ or $\mu^2 = \frac{\rho_0 e}{kT \varepsilon_0}$.
