Example: curl is 0 and/or divergence is non-zero for a field. Does that mean that it has this property at some point in that field or all the points in that field?
Like, I am thinking of a simple electric field lines due to a stationary charge. The divergence is non-zero at the point where the charge is located but if we move sufficiently far away then the lines are basically parallel and we have a situation where divergence is almost zero.
On
Classically both, as well as the gradient, are local operators. Applied to any smooth function they depend only on the values in the vicinity of a point. Generalizing to weak derivatives things are a bit different, with these derivatives being defined in terms of functionals, typically as integrations of test functions over a region with specific boundary behavior.
ADDED
But as you guess, and guess correctly you did, the classical derivative, curl, divergence can assign different values at different points.
The curl and divergence operators, $\nabla \times$ and $\nabla \cdot$, are operators which send scalar functions, say $f(x,y)$ to vector functions ($\nabla \times f)$ and scalar functions ($\nabla \cdot f$) respectively. So the curl and divergence are operators, which result in new functions, and therefore are global rather than local. So at two points in the field $f$, the divergence and curl will generally take two different values.