About having analytic control over any algorithm which finds perfect matchings.
Will it always return the same set of permutations in the same sequence?


No.  If you call the algorithm twice with the same input, it's not necessarily guaranteed to give the same output both times.  Many standard max-flow algorithms are non-deterministic, so there is no such guarantee.  For instance, consider any algorithm based on selecting augmenting paths.  If at any iteration there are multiple augmenting paths that would be acceptable, the max-flow algorithm will typically select one of them arbitrarily.  Consequently, it is not deterministic.

However, any such algorithm can easily be made to be deterministic by simply specifying a deterministic rule for resolving all non-deterministic choices.  For instance, whenever there are multiple acceptable choices, you can sort all of the acceptable options using some pre-specified order (say, lexicographic order) and always take the first.