How to prove that $X$ is a completion of $Y$, for some metric spcaes $(X,d)$ and $(Y,d)$?

To show that $\mathbb{R}$ is the Cauchy completion of $\mathbb{Q}$, it is not sufficient to show that $\mathbb{Q}$ is contained in $\mathbb{R}$ and $\mathbb{R}$ is complete. In fact, this just means that the completion of $\mathbb{Q}$ is contained within $\mathbb{R}$.

For example, the completion of $\mathbb{Q}$ is contained in $\mathbb{C}$ and the complex numbers are complete, but the completion of the rational numbers is not the complex numbers.

You should also verify that every element of $\mathbb{R}$ is the limit of a sequence of rational numbers (which follows directly from the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$ for example) in order to be sure that $\mathbb{R}$ is actually $\mathbb{Q}$'s completion. This resolves your worry about the uniqueness theorem for completions which you referred to.

One final point: one doesn't normally define the Cauchy completion in order to identify completions of metric spaces they already know (e.g. $\mathbb{Q}$ in your example). Instead, knowing that you can take the completion lets you construct new metric spaces which can be useful in their own right. Again for example, knowing that completions exist and taking the completion of $\mathbb{Q}$ is a way to construct the real numbers $\mathbb{R}$.