How Paillier cryptosystem can be used practically to encrypt and decrypt big messages "m"?
The Paillier cryptosystem allows to encrypt integers modulo $n$. Therefore, if $m$ is bigger than $n$, encrypting it will lose most of the message - only $m \bmod n$ is retrieved through decryption.

To encrypt a message bigger than $n$, you must break it into blocks, which you encrypt separately. You can for example write $m$ in base $n$, as $m = \sum_i m_i n^{i}$, and encrypt the $m_i$'s separately with Paillier.

Also, regarding how to calculate $g^m \bmod n^2$: note that $g = n+1$, hence

$g^m = (1+n)^m = 1 + n\cdot m \bmod n^2$

(if you develop $(1+n)^m$, you get $1 + nm + n^2\cdot \mathsf{something}$, and the $\mathsf{something}$ disappears modulo $n^2$).