Will any charge oscillating in space create an EM wave? Would it be correct to say that any charge oscillating in space (regardless of the spacial amplitude) at a given frequency will emit an EM wave of the same frequency?
related: What change in an EM field is required to create an EM wave?
 A: 
Will any charge oscillating in space create an EM wave?

Almost always yes, but a counterexample was given by G.A. Schott, "The Electromagnetic Field of a Moving Uniformly and Rigidly Electrified Sphere and its Radiationless Orbits," Phil Mag Suppl 15 (1933) 752. For a uniformly charged spherical shell of radius $b$, considering only retarded solutions, the radiation reaction force at time $t$ is proportional to $v(t-2b)-v(t)$. You get no radiation if there's periodic motion with period $2b/n$.
It doesn't seem to be possible to access Schott's paper online, but there is a 1964 paper by Goedecke that has a discussion of this type of thing. Schott was opposed to quantum mechanics, and his work is often cited by kooks, such as Randell Mills, who claims to have disproved quantum mechanics and discovered an unlimited source of energy. However, there appears to be nothing wrong with Schott's actual work in the 1933 paper.
Reading between the lines, Schott's hope seems to have been that he could salvage classical physics by showing that a hydrogen atom would not have to collapse by radiation, since it might be possible for the electron's orbit to satisfy this condition for not radiating. But this would only seem to work if the electron was a rigid, extended body orbiting at some speed that happened to be exactly matched to its own dimensions, and the electron also can't satisfy the condition unless the diameter of the orbit is less than the radius of the sphere.
A: It depends on the available space. In a free space - yes. In a perfect cavity the proper cavity modes are discrete and start from some minimal frequency. So a "slow" charge oscillations may not excite even the lowest cavity mode. The corresponding EMF is then purely reactive and may not propagate.
