Will the event horizons of a two black holes be perturbed or bent before a collision? What will the shape of the event horizon appear to be immediately after first contact?
Like del piero says, the gravitational field of each one affects the shape of the other, so it's similar to the "metaballs" used in computer graphics.
The exact shape will depend on the geometry, but it usually looks like two fingers reaching out to meet, like in Michelangelo's "The Creation of Adam". You can really see it in the first & last videos here, but there are some good videos of death spirals in between.
Note that the pulsing at the end of the first video isn't anything real; it's caused by the change in magnitude of the color scale shown in the bottom right corner.
This one has a shot of asymmetrically massed black holes, just after they join (with the original event horizons drawn underneath):
The apparent horizon is only affected to the point that the curvature around one black hole is modified by the gravitational field of the other.
The event horizon as defined by the boundary of the causal past of future conformal infinity gets modified in advance of significant changes in curvature, but this precisely what you would expect from a global teleological definition. See Hawking and Penrose for more details.