Suppose I have an open set $U$ in $\mathbb{R}^2$ and say it is diffeomorphic to $\Phi(U) \subseteq \mathbb{R}^2$ where $\Phi$ is a diffeomorphism.
1) Let $L$ be a path connecting two points $x$ and $y$ in $U$, and $L \subseteq U$. Does it then follow that $\Psi(L)$ is a path connecting $\Phi(x)$ and $\Phi(y)$?
2) Let $C$ be a non-self intersecting loop contained in $U$. Does it then follow that $\Phi(C)$ is also a non-self intersecting loop?
I guess I am just trying to understand what properties are preserved under diffeomorphism... Thank you.
Yes to both. In fact, you just need $\Phi$ to be any continuous map for this to be true, it need to be a diffeomorphism, or even a homeomorphism.
If $f:[0,1] \to U$ is a path with $f(0) = x$ and $f(1) = y$, then $\Phi \circ f:[0,1] \to \Phi(U)$ is a path from $\Phi(f(0)) = \Phi(x)$ to $\Phi(f(1)) = \Phi(y)$.
For (2), do the same composition "trick.'' A loop is just a special path, where the endpoints are the same.