Is SR applicable to phenomena in which objects accelerate?The earliest papers on SR considered accelerating electrons, so there is no good reason to exclude acceleration.
The answer is, of course, Yes. Special relativity would be useless if it were requiring all objects to move without any acceleration; after all, almost everything in the real world accelerates, otherwise the world would be useless. The correct claim similar to the proposition above is that special relativity has the same, simpler form in coordinate systems associated with non-accelerating observers. But that doesn't mean that we can't translate the predictions of a special relativistic theory to an accelerating frame. Yes, we can. It's as straightforward as a coordinate transformation. Fictitious forces will appear in the description. All of them are fully calculable.
Acceleration might be excluded because Einstein's 1905 paper had a section on kinematics, and that is sometimes regarded as the simplest and purest version of the theory. With that view, general relativity (GR) is the real theory, and SR is just a special and idealized case of little practical significance.
As Lubos explains, SR is the big theory. Its change to physics was broad and deep. GR is just the logical extension to gravity. GR is a lot more difficult mathematically, but most of the physics is in SR.