What is the relationship between the spin of a galaxy and the spin of its corresponding black hole?
Associated questions:
Do they always have the same axis of rotation?
Do they always spin in the same direction?
Does galaxy angular momentum increase due to frame dragging from black hole, causing continuing increase in galaxy angular velocity?
How does Hawking radiation impact rotational coupling?
We have to be careful and make a distinction between data and theory. First of all, as you can imagine from the name itself, a Black Hole (BH) is very hard to spot. It is even harder to measure its possible angular momentum (or charge, the quantities that, with the mass, characterize a classical BH). If we see evidences of a huge mass in a region of the sky, our General Relativity theory tells us that it has to be a BH. So there is a continuous exchange between model and data.
If we stick to pure data, at the best of my knowledge, we have no direct measure of the angular velocity of the internal black hole. At most, in almost all the cases, we can measure the speed of its accretion disk.
However, we can say something more in the particular case of Sagittarius A*. A particular case because it is a super-massive object lying at the center of our galaxy (so relatively close to us). In this case, hopefully soon, will be possible to test the angular momentum of the object via the "shadow" that a Kerr-metric solution predicts. In fact, a new instrument to study these BH features called Event Horizon Telescope is beginning to give results. So, Sagittarius A* is the best of our knowledge regarding BHs at the center of galaxies. Probably, what we don't understand here, we cannot understand somewhere else. You can find more information on a review by Tim Johannsen appeared in ArXiv today 1512.03818 (you can see how much of these is current research topic!! ).
Now, coming back to your questions,
What is the relationship between the spin of a galaxy and the spin of its corresponding black hole?
Experimentally, we don't know yet. Theoretically, it strongly depends on how the BHs born. There are many models of formation, I'm not an expert on supermassive BHs formation, and this is still an open question, it would be nice if somebody could fill this gap with some references on comments or another answer. Avoiding to fall into the "Chicken or the egg" dilemma, I can say that a BH is surrounded by its accretion disk, that in most of the cases spins. Once matter falls into the BH it will acquire angular momentum. The only way it could have some crazy (unrelated with the matter falling into) angular momentum it would be if some very strange and violent formation process occurred.
Do they always have the same axis of rotation?
Do they always spin in the same direction?
Does galaxy angular momentum increase due to frame dragging from black hole, causing continuing increase in galaxy angular velocity?
Obviously, frame dragging (FD) plays a role. But we cannot forget that all the matter drags spacetime. Even if the BH would spin in some crazy way, or stay fixed, its effect on the far stars it can be calculated negligible (I heard this, never tried the math thought). But, you can see its impact in the orbits of stars close to the BH (see 1512.03818).
Finally, your last question
How does Hawking radiation impact rotational coupling?
Hawking radiation is mostly a theoretical object, in astrophysics, it doesn't play any particular role. Especially, for those BH that are super-massive!!
Following Wikipedia,
$$T_h = {\hbar \, c^3 \over 8 \pi G M k_\text{B}} \;\quad \left(\approx {1.227 \times 10^{23}\; \text{kg} \over M}\; \text{K} = 6.169 \times 10^{-8}\; \text{K} \times {\text{M}_\odot \over M} \right)$$
e.g. the mass of Saggittarius A* is measured to be around $4\times 10^6 \text{M}_\odot $ getting an overall $T_{H}^{\text{Sag}} \sim 10^{-14} K$, this BH is in a thermal reservoir with temperature $T_{CMB}=2.7K$, so the black-body radiation of the Cosmic Microwave Background makes the Hawking radiation completely negligible. Remember that the Stefan–Boltzmann law tells you that the intensity of the radiation scale like $T^4$. You can see this related question to check how much the angular momentum does not change the game.
The definitive answer is that we don't know how the spins are connected. And, unfortunately, it might be a pretty long time before we have reliable observational evidence.
Black hole spins are very difficult to measure. Two methods have very recently started to be used, but both are approximate - and there is a lot of skepticism about the reliability of their results (especially since they often don't agree). Both methods rely on inferring how close the accretion disk comes to the black hole to infer the Innermost Stable Circular Orbit (ISCO) --- which then implies a spin magnitude and axis (but not which direction along that axis). The first method uses the temperature of the accretion disk (hotter means it comes closer to the BH), and the second method uses broadening of an iron emission line (the broader it is, the faster it is rotating and the more redshifted it is by strong-gravity -- thus the closer to the BH it is).
Both of these methods require fitting noisy data... The main uncertainty seems to be how well the inclination of the system (relative to the observer) can be determined. I.e. if the disk is edge on, could we distinguish that from face-on? How does that affect the measurement?
(Note that both of these methods have mostly been applied to stellar-mass BH systems, within our own galaxy. Massive Black-Holes in other galaxies is much harder.)
Even [super-]massive black-holes ([S]MBH) are very small compared to the sizes of galaxies they live in; like, 5-8 orders of magnitude smaller. This means it's very hard to simulate/calculate the connection between these size-scales. Specifically, we don't understand very well how MBHs get their gas from the surrounding galaxy. For that reason we don't have very good guesses on how galaxy and BH spins should be related. That being said, both simulations and people's theories suggest the spins should be correlated, but far from directly linked. Because of chaotic effects like galaxy and black-hole mergers, we can say with good certainty that in the 100's of billions of galaxies out there, lots of them will have completely opposite spins. We actually do know of some galaxies that have disk-like populations of stars which rotate in the opposite direction as the bulk of the galaxy... so this suggests the same could happen with BH accretion disks, and thus BH spins.
I can give a couple of definitive answers, however. Both frame-dragging and Hawking radiation from the black-hole definitely have zero effect on the galaxies spin, and evolution. Hawking radiation is completely negligible in all known astrophysical systems, and should only be relevant in micro-black-holes (if they exist[ed]). Frame-dragging is definitely an observable effect in the strong-gravity regime (near the event horizon), but because of the size-scale differences - it will have no way of coupling-to or affecting the galaxy as a whole.
Coupling between galaxy spin and central black hole spin
As far as I know, we have no evidence of any such coupling.
What is the relationship between the spin of a galaxy and the spin of its corresponding black hole?
As far as I know, We have no evidence of any such relationship.
Do they always have the same axis of rotation? Do they always spin in the same direction?
As far as I know, nobody knows.
Does galaxy angular momentum increase due to frame dragging from black hole, causing continuing increase in galaxy angular velocity?
As far as I know, no.
How does Hawking radiation impact rotational coupling?
As far as I know, it has no effect whatsoever*.
All very unsatisfactory I know. But there may be a sound reason for that. See this article: Monster Black Hole Spins at Half the Speed of Light. Note this:
"In the new study, a team led by Rubens Reis of the University of Michigan used NASA's Chandra X-ray Observatory and the European Space Agency's XMM-Newton — the largest X-ray space telescopes currently available — to observe the X-rays generated in the innermost regions of the disk of material circling and feeding the supermassive black hole that powers the quasar J1131. Measuring the radius of the disk allowed the astronomers to calculate the black hole's spin speed, which was almost half the speed of light".
Sounds good, doesn't it? But note that they haven't actually measured the black hole's spin speed. They've calculated it. And note this:
That's Einstein saying the speed of light varies with gravitational potential. Nowadays people tend to refer to the "coordinate" speed of light rather than just the speed of light. But it still leaves us with a problem, because at the event horizon, the coordinate speed of light is zero. The speed of light at that location, as measured by distant observers like us, is zero. And nothing can go faster than light. So what's the rotation speed of the black hole? What would we measure it to be? Saying it's rotating at half the speed of light doesn't help us when the speed of light is zero. And what does that do to conservation of angular momentum? Houston, we have a problem.
Yes, I know people take the Kerr metric for granted, but I have a nasty sneaking suspicion that the answers to your questions could all be negative because black holes don't spin. Because amazingly, incredibly, the ascending photon speeds up, and the descending photon slows down. To zero.
As for Hawking radiation, see Wikipedia where you can read about virtual particles popping into existence even though they only exist in the mathematics of the model, and about negative energy particles, which we have never ever seen. We have no evidence of such particles. And whilst we have evidence of black holes, we have no evidence of Hawing radiation. So my advice is to forget it.
