The speed of sound in a liquid is given by:
$$ v = \sqrt{\frac{K}{\rho}} $$
where $K$ is the bulk modulus and $\rho$ is the density. The bulk modulus of mercury is $2.85 \times 10^{10}\ \mathrm{Pa}$ and the density is $13534\ \mathrm{kg/m^3}$, so the equation gives $v = 1451\ \mathrm{m/s}$.
The speed of sound in solids is given by:
$$ v = \sqrt{\frac{K + \tfrac{4}{3}G}{\rho}} $$
where $K$ and $G$ are the bulk modulus and shear modulus respectively. The bulk modulus of iron is $1.7 \times 10^{11}\ \mathrm{Pa}$, the shear modulus is $8.2 \times 10^{10}\ \mathrm{Pa}$ and the density is $7874\ \mathrm{kg/m^3}$, so the equation gives $v = 5956\ \mathrm{m/s}$.
You give a slightly different figure for the speed of sound in iron, but the speed does depend on the shape and the figure you give, $5130\ \mathrm{m/s}$, is the speed in a long thin rod. There are more details in the Wikipedia article I've linked.
You can push the air faster than the speed of sound. If you do that, you will get a shock wave. A shock wave in this sense is a "wall" of supersonic-moving particles. You can definitely achieve this if you push on the air hard enough. A nuclear explosion is definitely "hard enough" :)
A shock wave will collide with "normal" stationary air, and give some of the energy to it. As the energy spreads to larger and larger volumes of air, the shock wave decays into a "normal" sound wave pulse. But before that, the wall of highly compressed air will travel at supersonic speed.
A nuclear fireball is the region where the air and the debris from the explosion is so hot that it glows. In the first moments of the explosion, the shock wave compresses the air so hard, that it heats up and glows. As the shock wave loses energy, it will lose it's glow, first "redshifting" and then disappearing almost completely. This phenomenon can be seen in this video: https://www.youtube.com/watch?v=KQp1ox-SdRI
Best Answer
I think what it is referring to is the difference between an oscillating pressure wave propagating through air molecules (like sound) and the air molecules themselves moving in one direction, without oscillation, and experiencing significant displacement (like wind).
The expanding gasses in a gun barrel, for example. The bullet is supersonic so the expanding gasses pushing it must be supersonic too, but that's not the same as the sonic boom you hear which only travels at the speed of sound.
EDIT: There is some vagueness of where the line is crossed between oscillation and displacement in my above description since it is pretty qualitative, but I found this answer: How do we get supersonic bullets?
which states the speed of sound increases with pressure, so if you have a high pressure front it will be able to move faster than sound at ambient.