Does real linear programming produce bipartite perfect matching using maxflow reduction? Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram:

We can formulate max flow as an linear programming problem  with integer variables in latter.


If we do not use integer variables does solving for maxflow in linear programming formulation with only real variables still produce valid perfect matching of given graph?

Is there a formal proof of this?
Let us consider $K_{2,2}$, the complete bipartite graph with two vertices on either side. A valid max flow sends $1/2$ units of flow across each edge of the bipartite graph. This gives a negative answer to your first question.

On the other hand, the integral flow theorem guarantees that there exists an integral max flow, and such a max flow can be found algorithmically. An integral max flow does correspond to a maximum matching.