I would claim that as it has matured into the modern psychoanalytic and cognitive theories James' stream of consciousness allows for a definition of the self, but does not require it. Actions are perceived as having agents, but those agents are not necessarily individual conscious beings.
There is a standard "social psychology" example of a crowd breaking into a riot based upon mutual observation of one another's growing emotionality, without any orchestration or purpose.
Who decides that a sports victory requires tearing down parts of buildings, destroying cars or trampling the elderly? Chicago and Edinborough somehow contain meta-beasts within them capable of causing that to happen. "We are so happy that too bad for you." (This is far from unique. I could have chosen Black Friday, but I am down on men today.)
This kind of thing is clearly an example of a thing with a narrative following a stream of impressions and acting. But that thing is hardly a 'self', even though there appears to be a 'will' that has decided there should be a riot, how it should spread, and how long it should continue.
That will expresses, if anything, more freedom than an individual. Any individual rioting mob member could choose to resist the action. He does not appear to be a mechanism driven deterministically by alcohol and testosterone, incapable of simply stopping. There is an ongoing re-engagement in the process that seems open to random change originating from any participant. And eventually, sanity takes hold of a critical mass of those involved and ends the process.
Theories of action that incorporate this perspective, have to see agents of many orders, which appear to have a composite will, but only some of which have a regulating mechanism akin to a self.
From this point of view:
1) No, a stream of consciousness does not require a self, and theories of psychological agency have to allow for conscious behavior involving both symbolism and emotion, that is not necessarily centered and guided by an established, unifying structure.
2) Yes, agency still requires an agent. Participation in action requires some attachment to the process or its goal. But that will does not need to be a cohesive and contained agenda that persists across time, it can arise almost randomly and assemble its agenda ad hoc, even if it provides a consistent motivation toward further action.
3) There is no reason to see this diffuse, animal will as being any less free than a centralized and planned one. It is as open to having random factors affect its agenda as any individual, and perhaps more so.
Going the opposite direction, toward Daniel Dennett's notion of "Brains made of Cells in Cells", we can see our own intelligence as arising out of a similar sort of mob action, with the freedom resulting from competition for resources within the brain, at a level below consciousness.
So the 'how' question is somewhat moot. Free will without a self, happens. And the two may even be inversely related. For traditional psychoanalysts, the self, with its sense of responsibility, may be a force that removes freedom from the will, for its own good.
If there is any non-determinism in the world, it probably infects competitive interactions most directly, as they concentrate differences that create advantages -- evolution runs on luck and it mines that luck out of every interaction.
If all of our emotional decisions are determined by some competitive interaction, it is wills with structure, not freedom, that are the harder thing to explain.