Using KVL for a mesh with a switch and inductor in parallel with capactior
I don't understand why you don't account for the voltage across the capacitor.


\$v\$ is the voltage across the capacitor. They've accounted for it by putting it in the left hand side of the equation. They just haven't written it in terms of the mesh current and constitutive relation of the capacitor yet.


  I am also a little confused why you cannot just take the derivative of the KCL equation like this ...


Your version has 

$$i_L = \frac{1}{L}\int v(t)dt$$

This isn't correct because it ignores the effect of the 6 ohm resistor.