Does inflation predict a multiverse? Does inflation necessarily predict a multiverse?
Or is it the case that "It all depends on the properties of the initial field responsible for inflation, about which we can only speculate"?
 A: forgive the shameless self-promotion:
My colleague and I, Michael Haslam,  studied an important version of this problem, specifically whether eternal inflation is a viable mechanism to produce a multiverse. We published two results:


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*In this paper , we showed that eternal inflation has serious mathematical issues, and most likely cannot produce a multiverse as serious singularities arise. 

*In this paper, we showed that even under more general geometric conditions, eternal inflation to be eternal must be extremely fine-tuned. As you suggested in your question, it is not only the initial conditions of the inflaton field, but also, how much initial anisotropy there is as well, for example. 
A: Regarding question 1, Inflation was offered as a theory to explain why the universe seems more or less the same and has the same cosmic background radiation in all directions today. A multiverse can be suggested from a multiplicity of theoretical positions. One could say there could be an infinite number, etc. One can argue, for example, given our small degree of understanding, it would be impossible NOT to assume there are multiple/parallel universes.
Re: #2,    I, for example, believe that scientists will sooner or later discover galaxies that  are 15,20,25 etc  billion yrs. old!  Then what? I believe "this universe" to be eternal and scientists will never, ever find the graviton, because gravity is an accelerating field dynamic understood by Einstein(gravity is not a force as the Standard Model begs it to be-it partakes in a geodesic).
Damn, I think I may have given you both more headaches and pondering. Right now the world of theoretical physics is a gigantic entangled ball of strings-pun intended. Too many egos running wild with whacky theories-among them the string babble. 
