Does the accelerating observer see radiation?
It depends on what you mean by radiation.
If you mean a travelling wave of oscillating electric field passing zero value twice per period, then no, there is no such oscillating electric field, because the charge does not perform oscillatory motion.
If you mean a time-dependent electric field that at any time, decays with distance from the charge as $1/r$, then yes, there is such a field in the accelerated frame; this however does not oscillate in time (or in position). We know similar situation from macroscopic EM theory, when e.g. very long solenoid increases its current - then, the electric field outside the solenoid (with circular line of force) decays with distance as $1/r$, but this is not called radiation, because magnetic field vanishes there.
If you mean that the charge produces an outgoing energy flow, then this is a more subtle question, whose answer depends on definition of EM energy we adopt. The usual definition meant for inertial frame - the Poynting definition - implies that a charge moving uniformly in an inertial frame does not produce any energy flow. In the accelerate frame, such particles accelerates, but the Poynting definition is not obviously justified there, so we can't apply it without further investigation of the question of how to define EM energy in accelerated frame.
The Poynting definition, in general, is not justified (and does not work consistently) for point charges. There are Frenkelian classical theories of point charged charges with appropriate EM energy definition which does work consistently. In these theories, accelerated point charged particles do radiate EM field that decays as $1/r$, but such field of one point particle is not associated with any EM energy flow, the field can propagate without energy propagating. To get and energy flow, one needs fields of two or more point charged particles. So in this class of theories, accelerated point particles does not produce an outgoing energy flow already in inertial frame. Again, this does not translate directly to non-inertial frames, but it illustrated that the energy flow question is more complicated.
From what I've read, though I'm not too sure, this is called "Unruh Radiation," and apparently there is no experimental evidence of Unruh Radiation. Doesn't this violate our notion of classical electrodynamics?
Unruh radiation is something different, it is a thermal radiation predicted from quantum theory to be present even if there is no charge. Yes, this violates classical electrodynamics, because in classical theory, if thermal radiation is present, this cannot be transformed away by change of frame.
Within the context of Maxwell's equations, a stationary charge viewed from an accelerating frame is indistinguishable from an accelerating charge in a stationary frame, correct?
No, it is incorrect. An accelerated frame is not an inertial frame, and we do not expect Maxwell's equations to hold in it.
