Are oscillating magnetic field sources always accompanied by an electric field? E.g. consider a rotating magnetic dipole.  Does the rotating dipole also generate a propagating electric field "disturbance"?
Basically, is the stereotypical in-phase, orthogonal electric/magnetic field propagation a special case that only occurs if generated under certain conditions, or does it occur for any oscillating electric or magnetic dipole (i.e. the only case)?
In antenna design, I thought I heard it was the case that "bad designs" won't propagate (e.g. if not properly impedance-matched to free space).
 A: Any changing magnetic field as well as any changing electric field, if not fully shielded, should produce some radiation. In that sense, a rotating magnet, which creates a changing magnetic field, will produce radiation.
The world is full of unintended antennas, but, I believe, you want to know what special conditions are needed to produce a strong radiation or to make a good antenna.
Basically, in a good antenna, the currents at different parts of the antenna should have such relative phases that the fields they produce add up (or constructively interfere) at a distance, at least in some direction. 
The outcome, in a given point at a given moment, will depend on the wavelength, the magnitude and direction of the current vectors and the distance from each current to the point in question.
If we take a twisted pair, the currents in the two wires of the pair are almost aligned with each other spatially and are out of phase, so we know that the fields they produce will mostly cancel each other regardless of the direction. Not a good antenna.

If, on the other hand, we take a simple tuned dipole, the currents in all parts of the dipole are aligned and are in phase with each other, which means that their fields at a distance will add up. This makes a good antenna.

However for the currents in a dipole to properly align, its length has to be about half of the wavelength of the transmitted signal. In this case, the current in the dipole will form a standing wave and, as a result, the currents in all parts of the dipole will move in phase. We can say that such dipole is tuned to or resonant at the transmitter frequency.
In another example, if we pass a low frequency AC current through a small loop (small loop antenna), the phase of the current in all sections of the loop will be apparently the same, but, for a current in any segment of the loop there will be a current flowing (vector pointing) in the opposite direction on the opposite side of the loop, and the fields, generated by these currents at a distance, will be subtracted from each other (interfere descructivley). Therefore, the radiation of such loop antenna will be weak even in the plane of the loop where the cancellation is minimal.
 
So, a small loop antenna will radiate much more than a twisted pair, but much less than a dipole.
If, however, the length of the loop is about one wavelength of the transmitter signal frequency (self resonant loop antenna), the loop current will form a standing wave, where the currents on the opposite sides of the loop will flow in the opposite directions in the wire, but - due to the geometry of the wire - their vectors will point in the same direction in space. As a result, the fields produced by these currents will add up in the direction normal to the plane of the loop, which lead to a strong radiation in this direction. 
 
This self resonant loop antenna is about as good as a dipole.
Of course, many real antennas are much much more sophisticated and are not easy to analyze, but, very often, their effectiveness comes down to synergistically combining the actions of many individual currents to maximize the total radiation in the selected directions. 
A: 
... consider a rotating magnetic dipole. Does the rotating dipole also generate a propagating electric field "disturbance"?

To prove your working thesis you have to place a wire near your changing magnetic field. If you do that you’ll find out for some directions of the wire in relation to the changing field that a displacement of electrons along the wire (an alternating current) could be observed. That is an empirical certainty, means it’s not necessary to repeat such experiments because it was done many times and the result is approved. So your working thesis is right and the answer is yes. In the presence of charges, a magnetic field will generate a current from these charges.

Basically, is the stereotypical in-phase, orthogonal electric/magnetic field propagation a special case that only occurs if generated under certain conditions, or does it occur for any oscillating electric or magnetic dipole (i.e. the only case)?

The changing current from an changing magnetic field has a sideways effect. The involved electrons get accelerated and as you perhaps know accelerating charges emit photons. Or in other words, they emit electromagnetic radiation. It was empirical find out - as you stated right - that both components of the electro-magnetic radiation oscillating perpendicular to each other. This is the case in free space. Under the influence of wave guides or external fields or lattice structures the two fields could oscillate under an oblique angle to each over. The orthogonal case is the most considered but not the only possible.

In antenna design, I thought I heard it was the case that "bad designs" won't propagate (e.g. if not properly impedance-matched to free space).

Any acceleration of charges produces EM radiation. The only possibility for an antenna rod not to emit radio waves is to disconnect the rod from the antenna generator. The generator accelerates the electrons forth and back inside the conductor. These electrons emit a lot of photons of different wavelengths during each acceleration cycle and what we measure is a swelling stream of these photons. It’s the carrier frequency of the radio wave. The receiver will filter out these frequency from the noise of all the EM radiation, means all solides, liquides and gases around us emit EM radiation, even an antenna rod with temperature higher the absolute zero. As long as a changing current can be induced inside an antenna rod, as long modulated EM radiation will be emitted.

Are oscillating magnetic field sources always accompanied by an electric field?

Additional to the above described effects, the rotation of any body, be this a conductor or an insulator, accelerates the charges inside the body and they emit EM radiation.
