Capacitance – Voltage characteristic diagram in a Schottky diode

In Britney Spears' Guide To Semiconductor Physics, on the page about pn-junctions, you can find this equation:

For your purposes, ignore the final version and just use the first and third parts of the equality

\$C = \dfrac{A e N_0 w_n}{V}\$

This equation comes from an assumption that the p- and n-sides of the device are equally and uniformly doped (\$N_{a} = N_{d}\$), and w_n is half the total depletion width.

If you don't have equal doping, you'll need to work out the depletion width for the n-side and p-side separately using equations (16) and (17) from Brtiney's page. This is not so bad in the relatively common case that one side is extremely highly (degenerately) doped --- which makes the depletion width on that side negligibly small.

The maximum rating on dv/dt is just that- do not exceed it if you want your design to be reliable. Older diodes were good for as little as 2kV/us but most are now okay with 10kV/us. Of course the actual ability to withstand dv/dt may be higher but that is all you are guaranteed, so stay within it.

The capacitance of all diodes varies with the voltage across it- minimum at the rated reverse voltage and higher with positive bias (though it is in parallel with the conduction of the diode). Diodes characterized for this are called varactor or varicap diodes, but all diodes have this characteristic.
Beware if this is important to you and the number is specified at a much higher voltage than where you are using it.

Schottky diodes do not directly have reverse recovery time as do conventional diodes (though the protective structures can have some similar effect), so the capacitance may be your main speed consideration. Here is the 1N5819 capacitance curve from the datasheet: