Do covariant derivatives commute?

It is false: take $M=\mathbb{R}^2$, $v(x,y)=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}$, $w(x,y)=\frac{\partial}{\partial x}$ and $f(x,y)=x^2+y^2$. Then: $$\nabla_vf(x,y)=-2yx+2xy=0$$ and thus $$\nabla_w(\nabla_vf)(x,y)=0,$$ but $$\nabla_wf(x,y)=2x$$ and $$\nabla_v(\nabla_wf)(x,y)=-2y\neq 0$$ for $y\neq 0$.

As you expected, this has nothing to do with a metric: the link between covariant derivative and usual derivation is, once you start with an affine connection on your manifold, it induces connections on all tensor bundles checking some properties regarding the connection you started with. One of these properties is that it coincides with usual derivation on the $(0,0)$-forms, which exactly says that on functions we have $\nabla_vf=vf=df(v)$ (pick your favorite notation here).