let $S_k(X)$ be the free abelian group with basis as the set of all k-simplexes . Prove that there exists a homomorphism $f_n : S_n(X) \rightarrow S_{n-1}(X)$ with $f_n \sigma = \sum_{i=0}^n(-1)^i\sigma(\epsilon_i^n) \in S_{n-1}(X)$, for any n-simplex $\sigma$.
If $g_n = f_n|_{\text{Basis of $S_n(X)$}}$ is the boundary of a singular n-simplex as defined above, then does it make sense that the homomorphism is $f_n : \sum_{\sigma}m_{\sigma}\sigma \mapsto \sum_{\sigma}m_{\sigma}g(\sigma)$ by extension of linearity? Or have I assumed something incorrectly here.
Yes, this is correct. Since $f_n$ is a homomorphism, we must have $$f_n\left(\sum_\sigma m_\sigma\sigma\right)=\sum_\sigma m_\sigma f_n(\sigma).$$ This is true for any homomorphism on an abelian group that is generated by elements "$\sigma$" (though in order to be sure that any such homomorphism exists at all, you need to know the group is freely generated).