Attached clipping from my lecture notes.
In this expression for the differential solid angle element I don't quite see how: $$ \sin\theta \, d\theta=d(\cos\theta) $$ Why is it not: $$ -\sin(\theta) \, d\theta=d(\cos\theta) $$ since the derivative of $\cos$ is $-\sin$.
It is indeed $-\sin(\theta)d\theta$, but you should also consider the integration limits. At $\theta=0$, $\cos(\theta)=1$, and at $\theta=\pi$, $\cos(\theta)=-1$. So $$\int_0^\pi\sin(\theta)d\theta=\int_1^{-1}(-1)d(\cos \theta)=\int_{-1}^1d(\cos \theta)$$