Concerning orientation: Is there any reason to think that the nearby stars rapidly orbiting give any hints? Or to assume that these stars formed along with the black hole and that way at some time shared momentum?
Yes, there are some subtle hints. Perhaps counter to intuition, the spin of the black hole has little to no relation to the orbital planes of those stars, or the galactic disk. This is because the black hole is the result of possibly many mergers over the life of the galaxy, and each merger involves adding the spin of another supermassive black hole with a random orientation. We've looked at the spins of several supermassive black holes in other galaxies (via their jets), and they are indeed fairly randomly oriented relative to their host galaxies.
We might instead wonder if the spin of the black hole shifts the orbits of nearby stars over time, through the frame dragging effect (Lense-Thirring precession). This effect is strongest very close to the black hole of course, and negligible for the galaxy, but the nearest of the S stars experience some measurable precession because of it. (There is also another precession caused by general relativity regardless of the spin -- the same effect that was noticed in the precession of Mercury's orbit around the Sun.)
There are two challenges, however. First is that each star only lets you place an upper bound to the spin and inclination. It's kind of like how when using the radial velocity method to detect exoplanets, if you don't know the inclination of the orbit, then you can only infer the minimum mass of the planet. So the method doesn't give you a definitive answer, but rather narrows down the possibilities. The second problem is that uncertainties in the measurements of the orbits and precessions of the stars can be significant. Still, this technique has been used
and was good enough to get some useful answers. (For example, it was able to rule out the spin axis being aimed directly at us.)
Many people have also wondered about the Fermi bubbles
, the two enormous x-ray and gamma emitting clouds above and below the galactic center. If we think of them as the remnants of black hole jets from enhanced activity in the galactic center millions of years ago, don't they imply that SgrA*'s spin axis is perpendicular to the galaxy, and hence we should see its accretion disk edge on? Clearly the EHT observation says no
, but we knew that reasoning was suspect anyway. Those bubbles may have little to do with jets at all. In fact they may have been a more omnidirectional outflow at first. The reason they extend more above and below the galactic plane is simply because those are the directions in which the density of the interstellar medium decreases most quickly. The bubbles follow the path of least resistance.