Min-cut in DAGs with unit edge weights
Your first claim is true.

As for the second consider the following counterexample. 
Consider the following DAG with edges $(s,v_1),(v_1,v_2), (v_1,v_3), (v_2,t), (v_3,t)$, where $s$ is the source and $t$ is the sink  

                 (v2)
                 /  \
                /    \
        (s)--&gt;(v1)   (t) 
                \    /
                 \  /
                 (v3)


There is only one mincut while according to your claim there are $\binom{5}{1} = 5$. So your second claim is false.

For the third claim consider the following DAG with edges $(s,v_1),(s,v_2), (v_2,v_3), (v_1,v_3), (v_3,t)$

                 (v2)
                 /  \
                /    \
              (s)   (v3)----&gt;(t) 
                \    /
                 \  /
                 (v1)


The Ford-Fulkerson (or any other) algorithm should find min-cut as the only edge $(v_3,t)$ which is not connected to the source. The the third claim is false.