First let me point out something about General Relativity: no black hole is created once you accelerate. You do get an event horizon for the accelerated observer, but that is not a black hole. A black hole is something defined globally, on the entire spacetime at once, while the accelerated observer is a local concept. You do get a behavior that resembles that of a black hole, but it is not a black hole.
Also, yes, you do get radiation and it is indeed called the Unruh effect. In short, the Unruh effect is the prediction that accelerated observers in Minkowski spacetime will perceive a thermal distribution of particles when an inertial observer would perceive none. Notice that both of them can be looking at the same thing at the same time: the thing is that even though the accelerating observer and the inertial observer can be at the same room, one of them will be freezing on vacuum (the inertial observer), while the other one will be burning with the thermal radiation (provided their acceleration is large enough).
The thing is that what both the Unruh effect and the Hawking effect are trying to tell you is that a "particle" is an observer-dependent notion. What one observer calls a particle does not need to coincide with what another observer calls a particle. That has to do with the fact that what one observer calls energy is not what another observer calls energy, which in turn has to do with the fact that time is relative (energy is just the conserved Noether charge associated with time translations).
That being said, let me try to address each of your questions in order.
Where did the radiation come from?
Different observers have different notions of particles. A static observer and an accelerating observer give the name "particle" to different things. They all agree on which state the quantum field underlying the "particles" are, but they disagree on how to interpret it. It is similar to how two observers might disagree on whether two given events are simultaneous or not in Special Relativity, albeit a little more complicated.
The radiation comes from the fact that once the observer is undergoing constant acceleration, what they call a particle is not the same thing they used to call a particle when they were static.
Where did the energy for the radiation come from?
From inertial forces. The thing is that the very definition of what is meant by energy changed once the observer started moving in an accelerated frame. Firstly, they were using energy defined according to inertial time. Later, they were using energy defined according to the time of an accelerated observer, which is a different concept. It is not equal, but similar to how a particle moving with constant velocity has different kinetic energies depending on the frame of reference: if you are standing still next to a ball of mass $m$, the ball has kinetic energy $0$, since it is at rest; if you start accelerating until you are moving at a speed $v$ relative to the ball, you now see the ball with kinetic energy $\frac{mv^2}{2}$. The ball's energy came from inertial forces, and that is due to the fact that you changed reference frames.
Has the radiation just disappeared?
Once the observer is inertial again, yes. Their definition of particle now coincides again with the one they had at the beginning and they no longer see particles laying around. The radiation is only observed while the observer is undergoing constant acceleration.
It might be a good point for me to mention that the transition steps between inertial movement, constant acceleration, and inertial movement are more complicated. I'm ignoring what happens in these bits (the notion of particle is way more complicated to discuss in them) and assuming we are always talking about the observers in stages in which they've been for quite a while, so the transition regions can be neglected.
Is this explained by known theories? (For example for a real black hole, there is no current theory which explains precisely how a black hole loses mass as it emits Hawking radiation).
Yes, and by "Yes" I'm also including the black hole evaporation issue. We can't really trust the evaporation computation until the end, but the matter of how a black hole loses mass is pretty much just a matter of conservation of energy (energy being defined in a very specific sense). These effects are quite standard calculations in Quantum Field Theory in Curved Spacetime and well described in any book on the subject (e.g., Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics). While experimental observation of these effects is extremely difficult, there are quite solid arguments in favor of them by means of consistency of usual Quantum Field Theory, consistency of Classical Electrodynamics, and experiments with analogue black holes.
Do we need quantum gravity to answer this question?
Not at all, or at least I've never seen someone bringing Quantum Gravity into issues involving the Unruh effect. The Hawking effect does have some difficulties due to extreme blueshifts that occur near black holes and the information loss controversy, but the Unruh effect is a quite solid prediction that is pretty much just as natural as regular computations using Quantum Field Theory, just in a non-inertial frame of reference.
Further Reading
In case you want to dive deeper into the Unruh effect and QFTCS in general, I suggest, in addition to the references I already linked, the following:
