I'm pretty far from an expert, but there are a lot of moving parts to this question and it's not easily answered, partially because we don't have a good definition of precisely what a black hole is. Special relativity tells us that it's a point singularity surrounded by an event horizon from which noting can escape. Quantum physics tells us that we don't know what it is, but hawking radiation can escape the event horizon and information is conserved on the outside . . . er, I think. A recent gravity wave discovery (see special relativity), indicates that they lose a pretty fair portion of mass when they merge.
To ship 1 it would appear that ship 2 would be moving slower to the
point it would seem like ship 2 stopped moving the closer to the black
hole ship 2 gets.
This is the common explanation, but I'd like to add some detail to it. This only works if the ship falls direct into the black hole with zero tangential velocity.
If we imagine a ship that's both reflecting visible light and radiating infrared light as it falls into a black hole, it speeds up very fast as it falls into the black hole, until the time dilation takes over and it appears to slow down, at which point it's very close to the black hole. It's velocity is a significant fraction of the speed of light - somebody else can do the math if they like, but If memory from another question is correct, about 1/3rd the speed of light when the slowing down takes over - to the observer. To the ship it continues to accelerate.
The ship continues to red-shift however, because the light that escapes the ship is red shifted both by the SHIP's increasing velocity combined with the gravitational red shift, so as a visual, you'd lose sight of the ship pretty quickly.
Something else happens because of a quirk with relativistic gravity. An object reflects and emits light in all directions, but as it gets close enough to the black hole, the only light that can escape is almost entirely perpendicular. Light can only escape the photonsphere of a black hole if it's close to perpendicular, and maybe my wording could use improvement there. This question has a very nice answer on orbital velocity around a black hole. Tangential light gets considerably bent or trapped entirely. What this means is you pretty quickly will lose sight of anything that isn't directly perpendicular between the ship, the center of the black hole and where you are viewing from. To actually continue to see the ship you'd need numerous cameras spread around the black hole and the view would need to be reconstructed.
And even if not for the photonsphere, the fact that parts of the ship would be closer to the event horizon than other parts, and that light would travel from the ship to your eyes at slightly different directions, you'd likely see some distortion, similar to the high velocity simulations you might have seen on relativity tv shows, or this website gives some examples, or when Randy Johnson throws a baseball.
And lets not forget, any object would likely break apart or spaghettify due to tidal forces, unless you had a really big black hole where spagetification wouldn't be an issue and the warping would be less but the photonsphere would still be a problem.
My point is, the story of the ship remaining visible is a nice simplification, in reality it's oversimplified. It ignores tangential velocity and it's frequently not mentioned that it would both spread out and quickly become so red shifted that it couldn't be recognized. The particle nature of light would suggest that in not too much time, the ship would be reduced to individual photons flickering and you'd need an enormous number of telescopes (assuming you could build a telescope to capture light that far red shifted), and an enormous amount of time to capture enough photons, to even track the ship at all. In practical terms it disappears pretty quickly. You might see it slow down and red shift and stretch for a while, but not very long.
And if the ship falls into the black hole with tangential velocity, well, that's even worse for keeping a visual track on it, and almost everything in space that falls into something else, has tangential velocity. The idea that the ship appears to freeze kind of in place and fade is probably not accurate at all but if any experts here want to correct me on that, I invite correction.
I wanted to point out this complexity before giving a layman's answer to your question.
The best way I can personally imagine what it would look like when two black holes fall into each other is to think of the photons outside the event horizon and what happens to them.
This is probably a pretty good estimate of what it looks like from ship 2. I think images like this are computer generated and not bad. One thing I disagree with is that these images are always drawn with a kind of ring around the black hole.
The black circle that you see is actually the photonshpere, not the event horizon and it's not completely black because some light with a tangential velocity can escape the photonsphere, but it's probably pretty dark, unless something very energetic is going on inside.
Also, those black holes above are drawn with a kind of light-circle around them. That's probably not accurate either. The entire space is warped, so there's no light circle. We imagine seeing a circle around an object that blocks a light because light can wrap around an object and we think of things as having edges, but that's not accurate. There would be warping around the photonsphere, extending outwards, but not a circle of light. I suspect there's a good chance you wouldn't see one black hole pass infront of the other, you'd just see it disappear, like black on black, then reappear as it warped space behind it when it was no longer infront of the other.
I don't object to the liberties taken to make videos like that more visible, but if you're asking what it would look like, I felt those corrections were worth pointing out.
Onto your questions:
because of the time dilatation is this gif picture below a good
rendering of how the merger would appear to ship 1?
(give posted above as well).
It's an OK rendering to give amateurs an idea of what it would look like. Time dilation is largely irrelevant because the black holes would orbit each other very very fast, unless they were very large black holes. That's another difficulty with your question. The answer changes somewhat with 2 supermassive black holes, like is expected to happen when Andromeda and the Milky Way's black holes are expected to merge, and with stellar mass black holes.
Stellar mass black holes would orbit each other very quickly and the time dilation would slow it down a little bit, but not enough where it would appear slow. The interesting stuff happens when they get very close. Take the collision that was detected by Ligo, when two roughly 30 solar mass black holes merged. A 30 solar mass black hole has a radius of about 90 km and if the two objects are orbiting each other close enough where you'd want to park a space ship and watch them collide, figure about 5 or 10 times that distance, say 900 km apart. Objects travelling at relativistic speeds around a (3.14 x 2 x 900), 5,652 km track, at relativistic speeds, you're looking at several orbits per second, give or take, that would likely speed up as they got closer. Time dilation doesn't catch up fast enough. You might get some red shifted slowed down bits and pieces, but like the space ship, the idea of watching the black holes kind of freeze on each other is probably not what it would actually look like.
A few other things to keep in mind. As the black holes spiral towards each other, they lose mass, so their event horizons actually shrink, which would be hard to observe until the very last moment where most of the gravitational wave energy is lost.
And, the black part is actually the photonsphere, not the event horizon. It could look to the viewer that the black holes are touching when it's just their photonspheres shaking hands, so to speak. If it's a small black hole and a bigger one, the small one could actually disappear into the larger one's photonsphere before the merger actually happens and the enormous pulse of gravitational wave energy is released.
So, the video probably isn't a very accurate representation of what it looks like, even allowing for the slow-down, but it's not a bad one either. I don't think it's meant to be accurate as much as it's meant to try to show what happens. It's like when they give Jupiter all these colors that it doesn't really have. It helps people like us "see" Jupiter, but an actual picture would be less colorful. It's OK to sacrifice some accuracy in order to make something more visual.
I imagine that 2 naked black hole(an infinity dense point with out a
surrounding black hole) merging would look like a blur from the speed
of them,
OK, here's where I'm in over my head, cause this is hard. I imagine that you're doing a thought experiment, not actually proposing "naked black holes", that is, if you could see the two singularities spiral towards each other, what would it look like.
First thing is, you can't see a singularity. Imagine you want to see one, so you sail your ship into a black hole. Sure it's a suicide mission but you will be the first person to see a "naked singularity" - WRONG. No light can travel from the theoretical point singularity outwards, so as you're inside the event horizon, not much changes. You see stuff that fell in with you, with different variations of tangential velocity. You see things stretched and warped by the curvature of space but you can't see the singularity because it doesn't send any photons in your direction. You could maybe observe some hawking radiation but even that is iffy, because unlike the border of the event horizon where one particle can escape and one can leave, inside, both particles would be drawn to the singularity and probably annihilate each other like good virtual particles are supposed to do. You'd see space, star light behind you, and stuff around you but you can never see the singularities (assuming that's what happens anyway, we don't have a good answer for what happens inside a black hole).
While you couldn't see it, you could model it. Two theoretical point singularities would, in a decaying orbit, spiral into each other. The gravitational waves released inside the black hole would be enormous, but, somehow, all that energy inside the black hole would likely remain trapped inside the black hole. The gravitational wave energy that's lost probably all happens before the theoretical singularity crosses the other black hole's event horizon, or before the event horizons touch and merge, so to speak.
But if you imagine there are two point singularities in this scenario, I see no reason why they wouldn't just very quickly spiral into each other. To get a better picture of this, we'd need an accepted model for quantum gravity, which we don't have. There are also models of black holes where the matter accrues on the outer edge of the event horizon, where time stops and there is no central singularity.
So the real answer to your question is that nobody knows what it would happen, but if we imagine two point singularities when two black holes merge, the laws of special relativity suggest that they would fly into each other very very fast, perhaps a tiny fraction of a second. Gravitational energy would be emitted as they spiral into each other, as defined by special relativity, but I don't imagine that energy would escape, it would remain inside the event horizon, but on that, I'm just speculating. There's many reasons why this question probably can't be neatly answered.
There's also a kerr black hole possibility where the point singularity is more of a rotating hoop or ring in order to retain it's angular momentum, but I've always had a hard time seeing how that meshes with losing energy via gravitational waves . . . but I'm not going to try to explain that one, cause it's above my pay grade.
Apologies if I waxed poetic a little too much, but I think this question almost works as a thought experiment, though it's probably impossible to answer.
