Question 1: A parameterization is not quite as strong as a(n inverse of a) global coordinate chart. For example, the curve in the line given by $|t|, -1< t < 1$ does not induce a coordinate chart, since it's not $1$-to-$1$. The image of this curve does, however, admit a global coordinate chart. The circle, on the other hand, does not, though it can be parameterized by $(\cos t,\sin t)$ with $0\leq t < 2\pi$.
Question 2: The classification of smooth $1$-manifolds already discussed does extend to topological $1$-manifolds. Specifically, the connected $1$-dimensional topological manifolds, up to homeomorphism:
- The circle $S^1$
- $\mathbb{R}$, or $(0,1)$
- The half-open interval, e.g. $[0,1)$
- The closed interval, e.g. $[0,1]$
I don't know of a detailed, published proof of this classification, but if you have access to JSTOR, here's an outline with copious hints.
Question 3: I don't think Lee means to equate curves with $1$-manifolds. One common definition is as follows:
A curve is a continuous map $f:I\to X$, where $I\subset \mathbb{R}$ is an interval and $X$ is any topological space.
The difference from $1$-manifolds is that curves may self-intersect, e.g. as the cuspidal cubic below. This is a manifold everywhere but the origin, but there it's homeomorphic to the cross.
There are also curves that aren't $1$-manifolds anywhere. Consider the space-filling curves of Hilbert and Peano sending $[0,1]$ onto $[0,1]^2$: the images of these maps gives curves, according to the above definition, that are $2$-manifolds!
The sort of curve that is a $1$-manifold is usually called an arc-the injective image of an interval.
Wolfram's definition is strange. Since they define their context as analytic geometry, I doubt they meant the domain should be an arbitrary $1$-dimensional topological space. That would permit curves to be disconnected, too, which they normally aren't.