An U(1) group element can be parameterized as $g=\exp(ic\xi)$ where $c$ is an arbitrary constant. In physics textbooks, I have found that the parameter space of $U(1)$ is referred to as $\mathbb{S}^1$. However, I have the following three questions regarding this.
1. How do I show that the parameter space of $\xi$ defines a circle? I mean why not $0\leq \xi<4\pi$ or even $-\infty<\xi<+\infty$? What forbids such values for $\xi$?
2. Even if $\xi$ is restricted to $0\leq \xi<2\pi$, does it necessarily define a circle? I mean, on any closed path on a plane $\xi$ is restricted to $0\leq \xi<2\pi$. Isn't it?
3. Is it necessarily a unit circle? If yes, what indicates that?
The phrase "the parameter space of $U(1)$" is ambiguous to me. I would be suspicious of that phrase without a proper mathematical definition.
Certainly it does not mean the space of $\xi$ values, as your guesses in items 1 and 2 seem to assume.
If you wanted to assign a letter variable as a "parameter", then you could assign a dependent variable to the expression $\text{exp}(i c \xi)$, say $$z = \text{exp}(i c \xi)) $$ Then you could observe that the set of values of $z$ --- the "image" of the function $\text{exp}(i c \xi)$ --- is indeed $\mathbb{S}^1$ which is the unit circle in the complex plane. That does give you a reasonable mathematical notion of what it means to say "the parameter space of $U(1)$ is $\mathbb{S}^1$".
Here's some important features to notice, in order to understand why $\xi$ itself is a bad parameter for the whole of $U(1)$.
First, the function $z = \text{exp}(i c \xi))$ is not one-to-one, in fact it is infinite-to-one, because of the periodicity of the exponential function. This is bad if what you want is a one-to-one global parameter.
You could also try to restrict to various $\xi$ subintervals, for instance $0 \le \xi \le \frac{2\pi}{c}$, and then the restricted function is almost but not quite one-to-one, because the endpoints $0$ and $\frac{2\pi}{c}$ have the same image.
You could repair the endpoint problem by restricting $\xi$ to the half open interval $0 \le \xi < \frac{2\pi}{c}$. Now you get a one-to-one and onto function from this half-open interval to $U(1)$. However this creates a topological problem, because that function has discontinuous inverse, and so is not a homeomorphism.
So, if you care about having a parameter which represents each point of $U(1)$ in a one-to-one fashion, and if you care that your parameter represent $U(1)$ with the correct topology --- that is, by a homeomorphism --- then there's really no way to repair this so that the independent variable $\xi$ is a good parameter for $U(1)$. You have to use a dependent variable like $z$.