Could two galaxies (one big and one small)intersect at a velocity to allow the smaller black hole to escape but not the galaxy around it?
Can “rogue” supermassive black holes be made this way?
I don't think they're "made" this way, but I think yes, they can be "made rogue" this way.
Could two galaxies (one big and one small) intersect at a velocity to allow the smaller black hole to escape but not the galaxy around it?
I think the answer is yes, but not for the standard reasons described by Peter and Mark. I say this because I like to think that I know how gravity works, see this essay. That's because I've read the Einstein digital papers. See the second paragraph here, where Einstein said "the curvature of light rays can only occur where the speed of light is spatially variable". A gravitational field is a place where what's nowadays called the "coordinate" speed of light varies, and because of this light curves.
The mechanism is in essence a refraction, hence this paper: Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime. We don't call it gravitational lensing for nothing. The light "veers" like a car veers when it encounters mud at the side of the road. See Professor Ned Wright’s Deflection and Delay of Light article for more. He doesn’t say the light is deflected because spacetime is curved. Instead he says this: “In a very real sense, the delay experienced by light passing a massive object is responsible for the deflection of the light. The figure below shows a bundle of rays passing the Sun at various distances”:
When you combine this with electron spin and the wave nature of matter, you can then appreciate why an electron falls down. You can then apply it to matter in general. However when you try to apply it to black holes, it just doesn't work. A black hole is a place where the "coordinate" speed of light is zero, and it isn't a dynamical "spinor". So you're left with no mechanism by which a black hole falls down.
This suggests that in your galactic collision, the smaller black hole would sail right on through the larger galaxy like a bullet through mist. As to whether this is right I can't be sure, because the interior constitution of a black hole remains an open question. But IMHO it's food for thought.
Edit: I've just noticed this answer where Rob Jeffries said "the orbital speed of the black hole components just before merger is greater than half the speed of light". The problem with that is that a gravitational field is a place where "the speed of light is spatially variable". We have hard scientific evidence of this, in that optical clocks go slower when they're lower. We see photons as blueshifted because we and our clocks go slower when we're at a lower gravitational potential. All in Einstein's general relativity creates some issues for contemporary black hole physics.
I assume you're asking about central supermassive black holes (SMBHs, one per galaxy), not stellar-mass black holes.
The answer is yes, but what actually happens is the two SMBHs have to merge first, and then the resulting combined SMBH can sometimes be ejected from the combined (merged) galaxy.
[Edited to add: Since you've updated the question with a series of diagrams, I should state explicitly that the scenario suggested by the diagrams -- stars in smaller galaxy merge into big galaxy, but SMBH continues on almost unaffected -- is not physically possible. Most of the stars from the smaller galaxy will not end up in the center of the big galaxy, but because of dynamical friction, the SMBH will.]
This NASA press release from 2017 describes the discovery of a quasar apparently ejected from a recently merged galaxy. I'll go ahead and quote their description of the suggested mechanism (this possibility has been suggested by theoretical studies going back at least ten or fifteen years):
According to their theory, two galaxies merge, and their black holes settle into the center of the newly formed elliptical galaxy. As the black holes whirl around each other, gravity waves are flung out like water from a lawn sprinkler. The hefty objects move closer to each other over time as they radiate away gravitational energy. If the two black holes do not have the same mass and rotation rate, they emit gravitational waves more strongly along one direction. When the two black holes collide, they stop producing gravitational waves. The newly merged black hole then recoils in the opposite direction of the strongest gravitational waves and shoots off like a rocket.
Since most massive galaxies -- including those that have undergone major mergers in the past -- have a SMBH in their center, the gravitational recoil usually isn't strong enough to eject the SMBH; instead, the SMBH loses energy to the stars in the inner part of the merged galaxy via dynamical friction, and settles back into the center. But it appears that sometimes there's enough of a kick to allow the SMBH to escape.
Another possibility is that if two galaxies merge and their SMBHs form a binary, and then another galaxy (with its own SMBH) merges before the previous two SMBHs have actually merged, then you can have a three-body interaction between the late-arrival SMBH and the binary SMBH, which could result in one of the SMBHs being ejected. But this requires the right timing, and probably doesn't happen very often.
Yes, and in fact a mechanism somewhat like this has probably dumped a large number of BHs into intergalactic space.
Black holes tend to settle towards the center of galaxies (an effect of dynamical friction). As they settle, they "cool" by evaporation. The chaos of BHs orbiting the center of mass all interact, especially when two of them approach closely. Depending on the geometry of the near-miss, one BH can gain energy at the expense of the other. One swings into a larger orbit and the other goes into a smaller orbit.
Sometime the larger orbit is hyperbolic and the BH is thrown right out of the galaxy. This removes orbital energy from the assemblage of BHs and the whole thing shrinks a bit and encounters become a bit more common. In the end, many of the original set of BHs are thrown out into intergalactic space.
How many? No one knows yet. We have good evidence of a single very large BH (>106 solar masses) at the center of the Milky Way, but recent results have suggested there may be as many as 10,000 smaller BHs (~10 solar masses each) in orbit around it.
If the latter is right, there may be a lot of BHs wandering intergalactic space!