Minimum-cut with minimum number of edges
Those answers assume that all edge capacities are integers.  Assuming they are, this works.

Suppose the min-cut in the original graph has total capacity $x$; then it will have total capacity $x(|E|+1)+k$ in the transformed graph, where $k$ counts the number of edges crossing that cut.  Note that if you consider any cut in the original graph with larger capacity, its original capacity will be $x+1$ or more, so its total capacity in the transformed graph will be $(x+1)(|E|+1)+k'$ for some $k'$ in the range $0 \le k' \le |E|$.  Now

$$\begin{align*}
(x+1)(|E|+1)+k'
&amp;= x(|E|+1)+|E|+1+k'\\
&amp;&gt; x(|E|+1)+|E|\\
&amp;\ge x(|E|+1)+k,
\end{align*}$$

so any cut in the original graph that isn't a min-cut won't be a min-cut in the transformed graph, either.

In other words, any min-cut of the transformed graph will be a min-cut of the original graph.

Moreover, out of all of the min-cuts in the original graph, the one whose cut set is smallest is the one that will have lowest capacity in the transformed graph (since the transformation adds one for each edge in the cut set).  So, the min-cut in the tranformed graph will be a min-cut in the original graph, and it will break ties by choosing the one with smaller cut set.