(2) is always true. However, (1) is only true on a simply connected domain. For instance, you can have a vortex with a singularity at the coordinate origin, and zero curl anywhere else. In that case, the function is not conservative, there is no potential, because you can pick up circulation around the "hole" in the middle.
Example:
$$\vec{v}(\vec{r})=\frac{(-y,x,0)}{x^2+y^2}$$
where I arbitrarily chose the $z$ axis to be the vortex axis.
p.s. yes, curl is defined as $\nabla \times \vec{v}$, which has no meaning for scalars. The same goes for the divergence. That's why it's much better to use $\nabla$ notation as opposed to spelled-out (and language dependent) names which don't allow math to be performed on it. For instance, $\nabla\cdot (\nabla \times\vec{v})=0$ is pretty obvious, but $\operatorname{div} \operatorname{curl} \vec{v}$ has to be remembered.
However, you can have more indices than one in these operations. The name divergence is also used, if only one tensorial index is consumed. Writing in the Einstein notation, you have
$$\operatorname{div} \vec{v}=\nabla_i v_i$$
but also on a rank-2 tensor:
$$\operatorname{div} {\sf A}=\nabla_i A_{ij}$$
For instance, force = divergence of the stress tensor.
Divergence therefore means contraction (summation) of the del operator with one of the indices of the differentiated expression, $\nabla_i A_{i(jkl...)}$. The same could go for the curl, in which case you would have $\epsilon_{ijk}\nabla_j A_{k(lmn...)}$. Notice the optional extra indices.
