Equivalence of following statements about shortest path problem We formulate the shortest path problem as follows:

We have a directed graph $D=(V,A)$ with length $c_{j}$ for each arrow $e_j$ in $A$ and two special points $s,t\in V$.
  The node-arc incidence matrix $A$ is defined as $a_{ij}$ is $1$ if the arrow $e_j$ leaves from $i$, $-1$ if the arrow $e_j$ arrives in $i$ or $0$ otherwise.
  Our decision variable is $f_j$, which is $1$ if the path uses arrow $e_j$.
  PRIMAL: We want to minimalize $\sum_{e_j\in A}c_jf_j$ under the conditions that $Af=(1,0,...,0,-1)^\top$ and $f\geq 0$.
  DUAL: We want to maximize $\pi_s-\pi_t$ under the conditions that $\pi^\top A\leq c$ and $\pi\in\mathbb{R}^m$.

I have the following three statements:
(i) There exists a shortest s-t walk.
(ii) The dual $LP$ problem has a feasible solution.
(iii) There is no circuit with negative total length.
Unfortunately I have absolutely no idea how to begin to prove the equivalence of these statements.
EDIT: Extra condition: from each vertex there exists at least one directed path to $t$.
 A: As in the comments above, you will need to make additional assumptions, such as assuming the graph is irreducible (your edited extra condition is not enough).  Here is a slightly less restrictive set of assumptions that also work:
Assume:
-For every vertex, there is a directed path from $s$ to that vertex.
-For every vertex, there is a directed path from that vertex to $t$.
-The lengths $c_j$ are real numbers (possibly negative, but certainly finite).
Solution approach to show equivalence between (i) and (iii):
a) Suppose the graph contains a cycle with negative length.  Show that for any arbitrarily large positive integer $M$, we can find a walk from $s$ to $t$ with total length less than $-M$.
b) Suppose there are no cycles with negative length.  Show that for every walk from $s$ to $t$, we can find another walk from $s$ to $t$ that has total length less than or equal to the first, and that does not repeat any nodes.  [Hence, to find a shortest walk, we can restrict attention to walks that have at most $N-1$ hops, where $N$ is the number of graph nodes.  That is, we can restrict attention to a finite number of options.]
