What is the final destiny of a neutron star? As I understand, neutron stars are born as extremely bright, extremely fast spinning cores of stars dying in a supernova. However, several websites tell me that within a course of a few years, the surface temperature of a neutron star falls from several trillion kelvin to only a few million kelvins. Furthermore, with the passage of time, the spinning speed of the neutron star also decreases considerably.

This raises the question: what is the final destiny of a neutron star? Does it always stay so horribly magnetic, hot and fast spinning or does it keep degrading into some form of cold, extremely dense star core with a much weaker magnetic field or do some of its features (specially the magnetic field strength and spin) stay at hightened levels forever (or at least several hundreds of billions of years)?
Neutron stars have extremely small heat capacities. That is because they consist largely of degenerate fermions and the heat capacity is further suppressed if, as expected, those fermions are in a superfluid state.

This has (at least) two consequences: 

(a) they cool down extremely rapidly - neutrino emission processes are highly effective, in the first $10^5$ years or so of a neutron star's life, at reducing its interior temperature to a few $10^7$ K and the surface temperature to $&lt;10^6$ K. After that, the dominant cooling process is photons emitted from the surface ($\propto T^4$) and neutron stars rapidly fade from view thereafter.

(b) However, the low heat capacity also means that it is easy to keep a neutron star hot if you have any way of adding energy to it - such as viscous dissipation of rotation by friction, accretion from the interstellar medium or ohmic heating by magnetic fields.

No isolated neutron star surfaces have been measured with temperatures much below $10^6$ K - i.e. all observed isolated neutron stars are at young ages. The situation is summarised in section 5.7 of Yakovlev &amp; Pethick (2004). Without any reheating, a neutron star would reach 100K in only a billion years - this is already utterly invisible. The reheating mechanisms must play some role for older neutron stars, but as Yakovlev &amp; Pethick state:  "Unfortunately, no reliable observational data on the thermal states of such stars are available". In conclusion, nobody really knows at the moment what the long-term ($&gt;10^6$ years) fate of neutron stars is in terms of their temperature.

The situation with regard to spin and magnetic field is more secure. There are not the same mechanisms available to spin-up an isolated neutron star or regenerate their magnetic fields. Both are expected to decay with time and indeed the spin-down rate and magnetic field strength are intimately connected, because the spin-down mechanism is the emission of magnetic dipole radiation. The magnetic field decays through the generation of currents that then ohmically dissipate (providing a source of heat) or perhaps more rapidly via currents generated by the Hall effect or through ambipolar diffusion.

For pure magnetic dipole radiation, one predicts $\dot{\Omega} \propto \Omega^3$. For typical surface magnetic field strengths of $10^8$ T, pulsars spin down to periods of around a few seconds in less than a million years, at which point the "pulsar activity" switches off and we can't see them any more, unless they are in binary systems and accreting matter in order to spin them up again. Unfortunately, there is very little observational evidence to pin down how fast magnetic fields decay (because we don't see old, isolated neutron stars!). The decay of B-field cannot be very fast, certainly timescales are longer than $10^5$ years. Theoretical estimates of B-field decay timescales are more like billions of years. If this theory is right then neutron stars would continue to spin down very rapidly even after the pulsar mechanism has ceased.