Superman can alter the timeline. The ability to change the past - present - future and know outside of the time continum what's happening and have the ability to change it I'd think would make any superhero instantly the most powerful.
Superman is also forbidden to interfere (a personal restriction at the urging of Jor-El, something highlighted in the first movie but ultimately ignored) with the timeline unless something has been done to alter the timeline - in which case he'll interfere to correct things.
I don't think he can alter the timeline as he chooses because he doesn't have an awareness of the continuum like you suggest. If he did, then he could simply correct every mistake and know the outcome to every decision before he has made it.
In the second movie - as it was intended before Richard Lester took over. Superman's turning back time in order to save Lois Lane and stop both missiles in the first movie, caused the release of the Kryptonian villains from the Phantom Zone. This was a consequence of his actions and even though this was later rendered moot by Superman turning back time again, (which I don't like because it suggests that he can turn back time whenever he feels like it) it showed that he was unaware of the chain of events that would follow an alteration in the timeline.
I do think that Superman is perfectly capable of turning back or travelling back in time, but I think that he avoids doing it because it's unpredictable and it breaks the rules that he's imposed upon himself.