In a way, yes. The "actions" are implicit in the underlying event space and dependency structures. In general definition of a stochastic process as a set of random variables indexed by an ordered set, the predictable part would come from our knowledge of the dependency structure and the particular sample path; the random part would be everything else we don't know about the sample path.
Yes - it depends on how the underlying event space is formulated. Take a coin toss as an example, where [tex]X_t=X_0[/tex] for t>0 and [tex]X_0[/tex] is a random variable that maps events to either 0 or 1. The underlying event space could be just the 2 events Heads or Tails, or it could be a description of the entire universe leading up to the point in time of the coin toss. The former model summarizes the uncertainties of the latter.
Hope this helps.