Classification of simply connected open sets of $\mathbb{R}^n $ modulo diffeomorphism Does that classification exists?
 A: The situation becomes much more complicated in higher dimensions. Already in dimension $3$ there are uncountably many simply connected (even contractible) non homeomorphic open subset of $\mathbb{R}^3$.
See for example this paper in which the following theorem is proved:

Theorem 2: There exist uncountably many contractible open subsets of $\mathbb{R}^3$, no two of which are homeomorphic. Hence, there are uncountably many different ways to express $\mathbb{R}^4$ as the product of a 3-manifold and a line.

This happen even restricting to contractible open subsets: if you consider simply connected open subsets there are even more examples, i.e. $\mathbb{R}^3\setminus\cup_i\ p_i$ where $p_i$ are points.
If you consider dimension $4$ the situation is even more critical, see for example small exotic $\mathbb{R}^4s$: in this case you have even open subsets which are homeomorphic to a ball, but which are not diffeomorphic.
A: I don't think so. Take as an example $\mathbb{R}^3 \setminus \{ 0\}$. This is simply connected, but it is not homeomorphic to $\mathbb{R}^3$.
