Can we "see" into a black hole using gravity? I believe the "no hair" theorem means all black holes settle down into a state only determined by a few parameters, typically listed as mass, charge and angular momentum.  But I don't think they can settle down instantly, which means the interior of the black hole may temporarily have an asymmetric mass/energy structure inside that may be observed from the outside using gravity.
If we had a huge massive black hole, say Milky Way mass, it would have a big radius, around a light year. Let’s assume it has no hair — it’s totally symmetric. We place plumb bobs on strings (let's call them pendulums) all around the black hole, a comfortable distance away. The symmetry means each pendulum points in a line through the common center of the black hole.
Now let’s drop in a more regular-sized black hole, say 100 solar masses. As it’s penetrating the horizon the situation is asymmetric. The pendulums wouldn’t point to the center of the big black hole anymore.
Would “no hair” require it to instantly becomes symmetric again once it penetrates? Isn’t it more likely it takes on the order of a year to become symmetric just because of the distances involved?
In other words, we can observe the interior structure of a black hole using gravity detectors.
 A: Yes.  You're exactly right, deviations from no-hair do occur for example after BH mergers --- and hints of the "quasi-normal" mode ("ringdown") were observed in the LIGO detection.  The no-hair theorem is constructed for a static, stationary BH (i.e. fully settled).  In general, deviations from no-hair (magnetic fields, asymmetry, etc) will be radiated away on the scale of the light-crossing (or 'dynamical') time --- just like you suggest.
A: We won't see inside the horizon, but we will see up to close to the horizon.
We will be able to see with much higher sensitivity when we deploy in a few years the eLISA gravitational observatory in space, with 3 satellites separated by 1 million Kms in a triad configuration.  See http://www.livingreviews.org/lrr-2013-7
The article also describes the many tests of strong gravity (but not quantum) that it will be capable of. 
That much better sensitivity will allow us to detect the gravitational radiation from higher order multipoles, besides the quadrupole radiation we saw in LIGO. That will allow us to match more detailed models/calculations and thus 'see' the 'shape' (i.e., the different multipole components) close to the horizon. We will see more of the details of the amplitude and time variations during the merger and ringdown. We will also be able to determine if the n-pole moments before merger really match that expected for 2 Kerr black holes, and how they change from the no hair to the hairy dynamical horizon till they settle down. So it'll also test the no hair theorem to an extent seeing possible variations from a Kerr black hole before the merger and in ringdown as it approaches Kerr. 
A: If you are using words like instantly and think it even means something in general relativity, then you need to learn more about general relativity.
In special relativity you already learn that simultaneity depends on frame.
Black holes form from collapsing matter. And the event horizon is a one way surface. Not because it is magic, but because it separates earlier events from later events.
Spacetime is 4d and particles go on curves in 4d in shapes where the curve has a future pointing tangent.
Any 4d curve that crosses the event horizon has events along the curve from before the curve hit the horizon and from after it hit the horizon. The event horizon is really separating those events into before and after.
So if a star gets dense and forms an event horizon, each part of that star has events from before and after that part crossed the horizon.
What we see is the events from before.
Including the center.
You can wait a year or a million years or $10^{400}$ years. You will still be getting new information about events at the center of that star but they will be from before the center of that star crossed the horizon.
You could even simulate this kind of thing yourself. Let's say you bought a building and told them they have to throw you (the new owner) a party at 5pm today according to the clocks inside. They might decide to make the clock inside run slower and slower. So you look in and it's 4pm outside and 4pm inside. And you ask someone to watch the clock and tell you when it reads 5pm.
And then later when it's 5pm by your watch outside you look in and notice every clock in the building has only advanced to 4:30pm.
And then later when it's 6pm by your watch outside you look in and notice every clock in the building has only advanced to 4:45pm.
And then later when it's 7pm by your watch outside you look in and notice every clock in the building has only advanced to 4:52:30pm.
And then later when it's 8pm by your watch outside you look in and notice every clock in the building has only advanced to 4:56:15pm.
And then later when it's 9pm by your watch outside you look in and notice every clock in the building has only advanced to 4:58:07:30pm.
Each time you wait an hour according to your clock. You see the clocks in the building advance. But they are advancing half as much as they did the last time. You will never ever see the clocks advance to 5pm.
That's what happens when you look at a black hole. You always see events from before the horizon. 
No matter how long you wait outside the building, you can see inside, you can see everywhere inside but the clocks always read from before 5pm. The inside of the star gets boring to the people outside. But you can see it.
You never ever ever see an event horizon form unless you cross it. And you don't see it change unless you cross it.
So you can see the interior of the stuff that forms the black hole. But you always see it from before the horizon formed. You never see the events from after. There might not even be an after as far as you know.
A: The asymmetry you observe will be the asymmetry of the configuration with the infalling matter - in this case, with the additional infalling black hole. It becomes more symmetric with time - a very short time, but not instantly.  
