I Does non-gravitational acceleration produce waves?

The short answer is "no", but the question itself appears to be based on a mistaken premise.

The "observational equivalence" is between being at rest relative to a rocket accelerating in flat spacetime (no gravity anywhere), and being at rest in a static gravitational field. In both of these cases, the acceleration being felt is not produced by "gravity". This is obvious in the flat spacetime case, since there is no gravity. But even in the curved spacetime case, being at rest in a gravitational field, what is holding you at rest is not "gravity" but whatever non-gravitational force is preventing you from freely falling. If you are standing on the Earth's surface, for example, the non-gravitational force of the Earth's surface pushing on you is what is holding you at rest.

In short, in both cases, the acceleration does not involve any change in the curvature of spacetime at all. And gravitational waves are changes in the curvature of spacetime that propagate at the speed of light. So if there is no change in the curvature of spacetime, there are no gravitational waves.

 

If we are talking about a mass that produces non-negligible spacetime curvature being accelerated, we are talking about a different situation than the OP was referring to. If we are talking about "gravity and acceleration being equivalent", we are talking about the Equivalence Principle, which only applies in a small local patch of spacetime in which curvature can be ignored. Also, strictly speaking, the EP only applies to "test objects", which are idealized objects that produce zero spacetime curvature themselves. That was the scenario I was assuming in my response to the OP.

 

No, they wouldn't--at least, not in the sense of "acceleration" that is applicable to your OP, which is proper acceleration. The two black holes in your scenario would be in free-fall orbits around each other, with zero proper acceleration. (At least, they would when they were far enough away from each other; as they got closer and eventually merged, things would get more complicated, but there would still not be any simple sense in which either one would be accelerating in the sense of the EP.)

The "acceleration" you are talking about now is coordinate acceleration, which is not an invariant (proper acceleration is an invariant), and can't be the cause of anything (because only invariants can cause something else).

No. See above.

 

PAllen is correct that, if we drop the assumption of the mass being a "test object", then accelerating it can produce gravitational waves. I was just pointing out that that is a different scenario than the one the OP was talking about.