Does the Shockley diode equation apply to Schottky diodes and non-silicon diodes?
The Shockley equation with minor modifications is what is used in SPICE simulations for forward current in silicon, Schottky, and LED diodes.

The parameter list is given as (from here):

The main difference between the different types (other than the number such as Is which is quite different for an LED) is the temperature dependence of the saturation current for Schottky diodes. It includes a resistance term.
The reverse bias treatment is different from the Shockley equation and includes the GMIN for voltages &lt; -5nV and the breakdown region exponential (the latter of which isn't covered by the Shockley equation). The GMIN business is to aid convergence of the diffeq solver rather than to improve accuracy.
Note: Vj (junction potential) and M (grading coefficient) are used to model the capacitance of the diode- M is important in reverse bias and Vj for forward bias.

As an example of an LED model, the LTspice model for Lumileds LXHL-BW02
(a 350mA power LED) has Is = 4.5E-20, N=2.6, Rs=0.85
In general you will not find the saturation current on a diode datasheet (you should not use the reverse leakage term), rather you would adjust IS and N to fit the forward V-I curve.