When the Empire fell, the NR did not automatically rule the entire galaxy. Many worlds chose not to sign up as member states or formed their own factions. I believe some worlds that were fanatically loyal to the Empire did not want to join the New Republic. That is their choice and I would assume the NR would have a tough time liberating worlds that did not want to be liberated! A lot of worlds benefited from Imperial rule. Therefore a small Imperial Remnant existed post Jakku. The NR did not pursue those Imperials who fled to the Unknown Regions because they didn't know where they were. The Unknown Regions were thought to be too treacherous to journey into as no safe navigational routes had been charted. The NR was unaware that the Emperor had found a route shortly before his death. The FO is a legitimate state. Worlds do not have to be part of the NR. The NR would only get involved in a conflict with the FO if the FO attacked a NR world. The Resistance was technically fighting an illegal war. Once Hosnian Prime was destroyed THEN the Republic could openly declare war - but it had no fleet.