Has a name been given to this constant? The most I've seen is calling it k, but is there another name for it, or any statement for understanding its meaning (such as how e0 can be called the "permittivity of vacuum")?
The constant k appears because of a mismatch of units in the SI system.
It is related to c^2, but powers of ten come in because of the mismatch.
Gaussian units, in which V=q/r with no constant, is used by most working physicists in their own work. epsilon0 has nothing to do with the permittivity of free space, although all elementary texts do call it that, never with an explanation because there is none. Most graduate texts still use Gaussian units.
Couloumbs law is a special case of Gauss law. The factor of 4[itex]\pi[/itex] is due to the integration of the electric field over the surface area of a sphere. In the more general Gauss law, only the factor of [itex]\epsilon_0[/itex] is present.
The value of [itex]\epsilon_0[/itex] depends solely on the system of units used. The three most common systems used these days are the SI system, the Gaussian (or cgs) system, and the Heaviside-Lorentz (HL) system.