Why do neutron stars have solid crusts? A long time ago I read that neutron stars have a solid crusts that are several orders of magnitude harder/stronger than alloys here on the Earth. So how is this possible ? 
A neutron star has a surface temperature of some 50,000 °K, so how can anything "solidify" at these temperatures ?
I understand that a solid is hard because of the chemical bonds and sometimes the crystals that form in the solid, so the only way that a star with a 50,000 °K can have a solid crust is if the matter there is solid because of other means, and that's because neither chemical bonds nor molecules can exist at these temperatures.
So how can a neutron star crust (and matter in general) become solid at these high temperatures where molecules and neutral atoms don't even exist ? And can this solid matter really reach strengths several order of magnitudes the strength of our alloys ?
 A: The neutron star crust is separated into outer and inner regions. The outer is a crust of neutron-rich nuclei surrounded by degenerate electrons. The inner is similar, but the nuclei are even more neutron-rich and there are degenerate neutrons too.
The (qualitative) answer to your question looks at the ratio of electrostatic (Coulomb) energy to the thermal energy of the ions in the crust.
$$\frac{E_c}{E_{th}} \simeq \frac{Z^2 e^2}{4\pi r_0 \epsilon kT},$$
where $T$ is the temperature, $Z$ is the atomic number of the nuclei and $r_0$ is a characteristic separation between the nuclei.
This ratio increases with: decreasing temperature, with decreasing nuclei separation (ie increasing density) and increasing atomic number. When it reaches some critical value, the plasma "freezes" into a crust, with the ions locked into some solid lattice. The same phenomenon occurs in the cores of white dwarfs at similar temperatures and densities, and the process has been "observed" to occur via asteroseismology.
So what is going on here, is that although the crust is hot ($10^{7}$ K would not be unreasonable actually), the densities ($10^{11}-10^{15}$ kg/m$^3$) are high enough to solidify the plasma.
This is of course not the whole story. At very high densities, when the neutrons drip out of the nuclei, one has to consider surface energy terms and ultimately the neutron fluid "dissolves" the crust at about $10^{16}$ kg/m$^3$, possibly via several bizarre "nuclear pasta" phases,eventually forming a fluid of neutrons, protons and electrons.
