Calculating number of equivalence classes where two points are equivalent if they can be joined by a continuous path.

It is impossible to have uncountably many equivalence classes. Note that each equivalence class is an open set, since balls are path-connected and so if $x\in G$ then any open ball around $x$ contained in $G$ is in the same equivalence class. Now any nonempty open subset of $\mathbb{R}^n$ contains an element of $\mathbb{Q}^n$, so each equivalence class must contain some element of $\mathbb{Q}^n$. Since $\mathbb{Q}^n$ is countable, there can be only countably many equivalence classes.

More generally, this argument applies with $\mathbb{R}^n$ replaced by any locally path-connected separable space.

As an open subset of $\mathbb R^n$ is the union of at most a countable number of open balls (centered on points with rational coordinates and having a rational radius), response 3. is the right one.