consider $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,y)=e^{x}siny$ then $\nabla (f)$ is $e^{z} :\mathbb{C} \rightarrow \mathbb{C}$ which image is not convex , not simply connected. So a negative answer to the second and third part of your question. regarding the first part I do not know the complet answer. But I can say only the following:
for every $V\in \mathbb{R}^{n}$, $\nabla f[U].V$ is a connected subset of $\mathbb{R}$, since it is well known that the partial derivatives satisfies Darboux theorem. Morover as a consequence of chain rulle $\nabla f.V$ is a partial derivative. So it is interesting to consider the following question:
Let $A$ be a subset of $R^{n}$ such that $A.V$ is connected for all $V$, does this implies that $A$ is connected?