I am fairly new to differential geometry and approaching it with a physics background (in the study of general relativity), as a result I'm having a few struggles with terminology etc, so please bear with me.
As far as I understand the statement that a manifold $M$ is locally homeomorphic to $\mathbb{R}^{n}$ means that any point $p$ in a given subset $U$ of a manifold can be mapped to a point in a subset $V$ of $\mathbb{R}^{n}$ by a homeomorphism, $\phi :U\subset M \rightarrow V\subset\mathbb{R}^{n}$. As these "coordinate maps" just label the points $p\in M$ we are free to choose any coordinates we like to suit the problem we are considering. Now, if we introduce a metric on the manifold such that there is an intrinsic notion of distance between two points in a given patch on the manifold, then the homeomorphisms from the manifold to $\mathbb{R}^{n}$ do not in general preserve the distance between points on the manifold, such that the "coordinate distance" in $\mathbb{R}^{n}$ does not correspond to the intrinsic distance between them on the manifold $M$. Is this all correct so far?
Now, my issue is that I've read that in general one cannot construct a Cartesian coordinate map (i.e. one that directly relates coordinate distance in $\mathbb{R}^{n}$ and intrinsic distance on $M$. The standard coordinate system for Euclidean, flat space) from a given patch on a manifold to a subset of $\mathbb{R}^{n}$. Is this because as, in general, the geometry of a given patch on a manifold will be non-Euclidean and so it will not be possible to construct a homeomorphism between this patch and a subset of $\mathbb{R}^{n}$ such the intrinsic distance (i.e. the metric) is preserved, which is what is required for a coordinate system to be Cartesian? I understand that given a sufficiently small neighbourhood around each point the geometry of a manifold is flat and thus Cartesian coordinate maps can presumably be constructed, however, in general one will want to consider much larger patches of a manifold (particularly in general relativity), so is the reasoning a gave above correct (at least heuristically)?
(1) Yes, it is more or less correct, just note that a metric is usually defined on a smooth manifold, which is a slightly richer structure.
(2) The simplest example is the standard 2-sphere (boundary of a $3$-ball with the usual metric). For any point of a sphere, some neighborhood is homeomorphic to a piece of $\mathbb{R}^2$ but this homeomorphisms never preserves distances. The reason is that, on the piece of $\mathbb{R}^2$, you have Pythagorean theorem that holds exactly, no matter how small the piece is. On the sphere, however, the Pythagorean theorem does not hold and the sum of angles in a triangle is never 180 degrees, no matter how small neighborhood of a point you take.
So, the statement that "given a sufficiently small neighbourhood around each point the geometry of a manifold is flat" is actually wrong, although it is a common mistake. It is only true if the Riemannian curvature is zero in every point of the given neighborhood. The topology is locally Euclidean but the geometry is not.