Does minimum cost flow problem work for real valued edge weights/costs?
In the case of the Ford-Fulkerson algorithm, yes, it's due to complexity. When the edge weights/capacities aren't integers, things get really annoying, really fast. 

If you allow all real numbers, you can end up with an instance that doesn't terminate. Normally, you augment paths until you can no longer do so. An irrational capacity value could allow the algorithm to continually find smaller and smaller augmenting paths.

Uri Zwick came up with a good example to show this:


  
  
  Consider the flow network shown on the right, with source $s$, sink
  $t$, capacities of edges $e_1$, $e_2$ and $e_3$ respectively $1$,
  $r=(\sqrt{5}-1)/2$ and $1$ and the capacity of all other edges some
  integer $M \ge 2$. The constant $r$ was chosen so, that $r^2 = 1 - r$.
  We use augmenting paths according to the following table, where $p_1 =
&gt; \{ s, v_4, v_3, v_2, v_1, t \}$, $p_2 = \{ s, v_2, v_3, v_4, t \}$ and
  $p_3 = \{ s, v_1, v_2, v_3, t \}$.
  
  
  
  Note that after step 1 as well as after step 5, the residual
  capacities of edges $e_1$, $e_2$ and $e_3$ are in the form $r^n$,
  $r^{n+1}$ and $0$, respectively, for some $n \in \mathbb{N}$. This
  means that we can use augmenting paths $p_1$, $p_2$, $p_1$ and $p_3$
  infinitely many times and residual capacities of these edges will
  always be in the same form. Total flow in the network after step 5 is
  $1 + 2(r^1 + r^2)$. If we continue to use augmenting paths as above,
  the total flow converges to $\textstyle 1 + 2\sum_{i=1}^\infty r^i = 3
&gt; + 2r$, while the maximum flow is $2M + 1$. In this case, the algorithm never terminates and the flow doesn't even converge to the maximum
  flow.


(Wikipedia)

The Stoer-Wagner algorithm will work with non-integer capacities, and so will Simplex/LP algorithms.