Your picture of a black hole is not an accurate one. Several issues:
(1) The "black hole" is not just the singularity. The term is used to refer to the entire region of spacetime inside the event horizon. When people talk about two black holes merging, they are talking about two regions inside event horizons merging into one region inside an event horizon.
(Strictly speaking, there is only a single event horizon, and only a single region of spacetime inside it; that region just happens to be shaped like a pair of trousers instead of a tube, so to speak.)
(2) A black hole is not "infinitely dense". The singularity itself can be thought of as "infinitely dense", but the singularity has no causal effect on anything else in the spacetime, so its characteristics are irrelevant for understanding what happens elsewhere.
(Strictly speaking, the singularity is not even "in" the spacetime--the spacetime itself "ends" at the singularity, meaning there are events arbitrarily close to the singularity but none actually "at" it.)
(3) The event horizon is defined "teleologically"--it is the boundary of the region of the spacetime (as above, there is only *one* such region, but it may be shaped like a pair of trousers instead of a tube) that cannot send light signals to "infinity" (strictly speaking, to "future null infinity"). That definition requires you to know the entire history of the spacetime to pin down exactly where the horizon is. So when an object of non-negligible mass falls into a black hole, the horizon starts to move outward from its old radius to its new radius even *before* the infalling object reaches it, because the horizon is defined in terms of where light signals go all the way into the infinite future. A light signal sent from outside the "old" horizon radius may still be trapped behind the new horizon even if it is sent *before* the infalling object reaches the "new" horizon radius--if it is sent a short enough time before, so that it doesn't have time to make it past the new horizon radius before the infalling object arrives.
Take a look at the diagrams on this page:
Particularly the Kruskal and Penrose diagrams of the star collapsing to a black hole. It may help to visualize what I'm saying above.