Two-dimensional S-Box
I understand the question as you have a single 4-bit S-box, which you first apply rowwise, and then columnwise. 

As already mentioned, this is equivalent to a large S-box $\mathcal{S}$
$$ 
c = \mathcal{S}(m\oplus k_1)\oplus k_2.
$$

This is a well-known Even-Mansour cipher, and it can be broken with complexity $2^{n/2}$, which is $2^8$ for your $n=16$. The idea by Daemen is to try $2^{n/2}$ plaintext pairs with the same difference $\Delta$ both for the entire cipher $E$ and the S-box $\mathcal{S}$.

So the answers to your questions are


No advantages.
Breakable in seconds.
Maybe in some marginal ciphers or hash functions. Note that it is quite inefficient in software.