Ford-Fulkerson vs Edmonds-Karp
Edmonds-Karp is a specialisation/elaboration of Ford-Fulkerson, so any bound for the latter also applies to the former.  In other words, EK is $O(|E|\min(f_{max}, |V||E|))$ time (and writing it this way does add information, since $f_{max}$ can be much smaller than $|V||E|$ -- and this is the only time when you might otherwise consider using some other variant of FF in preference to EK).

The above does not imply that EK is faster than a particular (say, DFS-based) variant of FF on any particular instance.

BFS and DFS have the same runtime, but DFS only promises to find a path from source to sink in the residual graph -- not necessarily a shortest possible such path, which BFS does promise.  It's possible to show that no modification to the residual graph ever reduces the distance from the source to any vertex; with this, and the fact that BFS always finds a shortest path, it's possible to show that after finding at most $|V||E|$ augmenting paths, no source-to-sink path remains in the residual graph.  This page gives a full proof -- I've only skimmed it, but note that when it says that "That is, [the distance from the source to all other nodes] is always increasing", it in fact means (or should mean) that this never decreases -- that is, the distance from the source to a particular node can stay the same after an augmentation, or it can be larger, but it can never be smaller.