Are gravitational waves produced only by masses which experience a change in acceleration, and if so, is this connected to the spin 2 of the graviton? I can't remember for sure, which is why I ask this question, but is it the case that gravitational waves are produced not by accelerated masses (like e.m. waves are produced by accelerated charges), but only by masses with a change in their acceleration?
And if so, is there a connection with the conjecture that the (hypothetical) graviton has spin 2?
 A: The answer is basically yes, though there's some arguments you first have to go through. It's a perceptive question.  
The important theorem which lays the groundwork for your claim is an important one by Weinberg. It is that in QFT, in a rather generic sense, a spin 2 particle (i.e., the graviton) must be the quanta for a gravitational interaction, which is simply one that obeys the equivalence principle, i.e., all particles fall the same way in gravity, or stated differently, the acceleration of a particle in gravity is the same regardless of its mass or any of its properties. This was proven by Weinberg, similarly to the way that spin 1 photons imply charge conservation. See a nice discussion and derivation in a Columbia U. set of of notes at http://phys.columbia.edu/~nicolis/GR_from_LI_2.pdf. They have the Weinberg references. 
The only next step is to understand that the equivalence principle leads you to General Relativity. The equivalence principle also implies that gravitational free fall and a uniformly accelerated reference frame are indistinguishable, in a small enough neighborhood or lab. A uniform  acceleration is just geodesic motion, and does not radiate gravitational waves. You have to change the acceleration to create gravitational waves. Yes, you need more than acceleration of a single particle to radiate gravitationally
Now, to be clear, there's lots of ways to get gravitational radiation, and lots of ways not to, in General Relativity, all in accord with the equivalence principle. Spherical or axial symmetry cause no gravitational radiation, even with high rotation rates. You could get gravitational radiation from 2 particles (or black holes or planets) orbiting around each other, there is enough of an asymmetry. THis non local distribution of matter means there will be varying accelerations. Anything like that has a quadrupole moment or higher moments, and gravitational radiation is produced by second time derivatives of the quadrupole moment (that is the strain is, and the tidal field is the fourth time derivative), or higher derivatives of higher multipole moments. 
