Length of finitely generated module over $0$-dimensional Gorenstein local ring

Please see attached images: Can someone slowly walk me through the details of the proof of Lemma 4.1 using basic commutative algebraic arguments and definitions?
1) Why $R[\mathfrak{m}] \cong k$? (where $k$ is the residue field of $R$)
2) I know length is additive on direct sums, but how is the RHS of the inequality additive as well, since it is a product?
3) Why does the inequality coincide when $R$ is the module over itself?
4) How does $R$ being Gorenstein of dimension $0$ imply that $R$ is the only indecomposable injective module, and why is its injective hull free as a result?
5) Why $M \subseteq \mathfrak{m}F$?
6) Why does $M=0$ imply the desired result?