For the definition of Schwartz space space on $\mathbb R^{n},$ see this.
My Questions:
(1)Is it make sense to talk of Schwartz space on torus $\mathbb T^{n}$ ? If yes, what can be the analogous definition of Schwartz spaces on torus $\mathbb T^{n}$ ?
(2) What are the differences between $\mathcal{S}(\mathbb T^{n})$ and $\mathcal{S}(\mathbb R^{n})$ ? (For instance Fourier transform maps $\mathcal{S}(\mathbb R^{n})$ onto it self; what about $\mathcal{S}(\mathbb T^{n})$ ? etc...)
(3) What are tempered distributions on $\mathbb T^{n}$ ?
There is no Schwartz space, just $C^\infty(\mathbb T^n)$. Indeed, $\mathcal S(\mathbb R^n)$ is an intermediate class between all smooth functions and compactly supported smooth functions. But on $\mathbb T^n$, everything is compactly supported.
Since the Fourier transform of a function on $\mathbb T^n$ is a function on $\mathbb Z^n$, it does not make sense to talk about a class being preserved under Fourier transform.
Since there is no distinction of "quickly decaying vs compact" smooth functions, there is not a separate concept of tempered distributions. You just have ordinary distributions. They can be characterized in terms of Fourier coefficients: since the Fourier transforms of $C^\infty$ functions are precisely those $g:\mathbb Z^n\to\mathbb C$ that decay faster than any polynomial, a distribution has Fourier coefficient that are bounded by some polynomial. For details, see this answer by Paul Garrett, and the notes linked from it.