Why do we require radians in calculus? I think this is just something I've grown used to but can't remember any proof.
When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?

In Calculus, sine and cosine are defined via the exponential function, meaning that 
$ \cos x = \mathrm{Re}\{e^{ix}\}$  and $ \sin x = \mathrm{Im}\{e^{ix}\}$ and as you know, $e^0 = e^{i2\pi} = 1$ which means that $360^{\circ}$ which is the full circle corresponds to $2\pi$. For further reference see here.
