1)
\$J_s\$ is a surface current density in units of amps per meter (\$A/m\$), as you said. This means that it is a "sheet" of electric current that flows parallel to the boundary between the two media. Its direction is determined by the formula you posted:
$$
J_s = n_{12}\times(H_2 - H_1)
$$
Which states that:
The tangential components of the magnetic field across an interface, along which there exists a surface electric current density \$J_s\$ (A/m), are discontinuous by an amount equal to the electric current density.
Where \$n_{12}\$ is the unit vector in the direction from media 1 to media 2, in this case the same direction as \$H_n\$ in the image below. \$H_1\$ and \$H_2\$ represent the tangential components of the magnetic fields in both media 1 and 2 respectively, for example \$H_t\$ in the image. If you take the cross product of \$n_{12}\$ and \$H_t\$ you will get a vector coming out of the page which is \$J_s\$. If \$H_t\$ is uniform from the bottom to the top of the figure, then we would expect \$J_s\$ to also be a uniform sheet of current, that also extends from bottom to top, flowing towards us(out of the page) along the interface of the two media (we assume that the material is three dimensional and extends into the page). If you were to calculate \$J_s\$ via this formula and you wanted to know the total current flowing along the boundary you would simply multiply \$J_s\$ by the height of the boundary (length from the bottom of the figure to the top) which would give you the total current in units of Amps.
2)
This relation holds in the case of any media. If both media have finite conductivity and there are free surface charges, as in a conductor, then this relation would describe the discontinuity between the magnetic fields as being equal to the surface current density. If there is no free charge, as in a perfect dielectric, then this relation reduces to:
$$
J_s = n_{12}\times(H_2 - H_1) = 0
$$
since the tangential component of the electric and magnetic fields must be continuous across the boundary and therefore \$H_1 = H_2\$.
If either of the media are a perfect electric conductor (PEC) then the tangential electric, and therefore magnetic, field in that material will be zero. For example if material 1 was a PEC then \$H_1 = 0\$ and now:
$$
J_s = n_{12}\times H_2
$$
In the example, the core permeability is assumed infinite (it says it in two places). With infinite permeability comes infinite inductance and therefore any reasonable AC excitation (frequency greater than zero Hz) will not cause any current to flow hence ampere turns or MMF and H (ampere turns per metre) have to be zero.
Further down they discuss two real materials that clearly don't have infinite permeability.
Best Answer
Math
As to why this is allowed: They follow from some of the math. When you solve the equations (you can actually get both of Gauss' laws from Faraday's and Ampere's laws), you end up with a constant, which is the charge. Think of it this way: Gauss' laws state that the total amount flow of field into/out a sphere equals the amount of 'sinking/sourcing power' inside of that sphere. The left part of the equation, the \$ \overrightarrow{\nabla} \cdot \overrightarrow{E}\$ and \$\overrightarrow{\nabla} \cdot \overrightarrow{B}\$ are that 'total flow into/out of the sphere'. The \$ \frac{\rho_e}{e_0} \$ and \$ \mu_0\rho_m \$ are the 'sink/source power'.
So from this mathematical perspective, it makes sense to write them. Just because we have not found them (and thus, the \$\rho_m\$ will always be zero), doesn't mean it is wrong to write them. If they do exist, it would make a lot of sense for the 'full' equation to be correct.
Duality
In addition to this, there is another reason we might want to write them down: There is a concept in electromagnetism, which is called 'Duality' (link to wikipedia page). I won't list all of the 'properties' here, but in short, this concepts states that you can exchange 'magnetic' and 'electric' concepts, and still keep valid equations (only if you exchange all the concepts - you can't choose just a few).
But, there is a problem with the electric charge - in the standard set of Maxwell equations, there is nothing to exchange with electric charge. This is where magnetic monopoles come in - we can exchange them with the electric charge and use this duality.
What is the use of this? It can be very effective for the design of new structures. Take for example the simple electrical dipole antenna we know and love. If we apply the duality principle to this antenna, we know we would have an antenna that performs the same, but would be a magnetic loop instead. And this is exactly what a loop antenna is - the magnetic dual to the electric dipole. The same applies to making more complicated structures.