Do most physicists believe that axions exist? Is the theoretical justification behind proposing axions sufficiently robust, like, say the Higgs boson prior to its detection, that one should consider the existence of axions as generally accepted (though unconfirmed) physics, or are they more speculative than that?
 A: Beliefs have no real place in physics, at least should not have. The proposal of particles not seen comes from the theoretical models in order to complete them. For example the omega minus was predicted more or less exactly from the symmetries of the weak SU(3), and was later found .A more exact prediction than the Higgs.
The Higgs as a particle might not have existed, as a higgs mechanism could be used to the same effect for symmetry breaking without biasing the theory  In this sense it was a lucky guess that it existed.
Axions can appear in a theoretical model , but they are necessary only for specific ones, as for example in this supersymmetric model where the supersymmetric partner of of the axion is necessary if dark mass can come out of the model naturally,

Because the light stau is the lightest supersymmetric particle (LSP) in the no-scale MSSM, to preserve R parity, we introduce a non-thermally generated axino as the LSP dark matter candidate.

So what you term "belief" is the degree of necessity for a particular model of nature. The broader the constraints, the more models there exist and "belief" gets lower.  Though one of these models may get lucky and fit future data.
