I read that a vector field has a stream function, only if it is conservative. And I also red that all conservative fields have a potential function. Is that right? I mean it can happen that: ∇⋅u=0 and ∇×u≠0? Like for instance:
f = cos(x)sin(y) i - sin(x)cos(y) j has a stream function, because ∇⋅u = 0 (∂v/∂y + ∂u/∂x = -sin(x)sin(y) + sin(x)sin(y) = 0).
But ∂v/∂y = cos(x)cos(y) ≠ ∂u/∂x = -cos(x)cos(y), and they would have to be equal if that vector field would have a potential function.
Your example answers your second question: it is of course possible that $\nabla \cdot u = 0$ and $\nabla \times u \neq 0$ for some vector field $u$, these are completely independent properties. Now for your first question: a vector field can have a potential only if it is curl-free, that is $\nabla \times u =0$. But the other direction is only true for simply connected subsets.