The Romans in general did not like following someone else's commands. They liked it so little that they had (as you note) a specific word for the right to give commands: imperium, the legal authority to give commands, or imperare.

Imperium, however, is carefully limited by one critical factor: it is only valid outside the city. In practice, this means on military campaign, so that a general had the power of command -- and the power to enforce his commands with punishment -- when leading troops afield, but didn't have that power when inside the city, which was presumed to be a time of peace. As a sign of this, consuls had the full fasces with an axe carried before them when abroad, symbolizing their power to punish with death, but when they entered the city the axe was removed.

Correspondingly, there was a strong link between military authority and imperium. Thus the word imperator, from impero, probably originally simply meaning "one who commands" -- that is, if we draw out the connotations, one who has the right and authority to command and to enforce his commands. Since imperium is almost exclusively exercised abroad by generals, it became a word applied as an appellation of generals. In its original signification, indeed, it was nothing more than a word for "general," and was equally applied to Roman commanders and the commanders of foreign armies.

But the Romans like titles. Specifically, they like adding titles to someone's name to create über-names. A general who had fought a war and won a great battle in Africa, for instance, could be surnamed Africanus, as in Lucius Cornelius Scipio Aemilianus Africanus; one who had triumped over the Numidians could be Numidicus; and so on. These surnames were sometimes spontaneously given by the soldiers on the field of battle (and sometimes granted or confirmed by the Senate).

Thus, when it became the habit of soldiers to start chanting Imperator, Imperator! to their general after a major victory, this took on a larger signification. Originally, of course, such a chant meant nothing more than General, general! -- but the victorious general, reflecting that a glorious title after your name is a nice thing to have, not unreasonably decided that being saluted imperator in the field, spontaneously, by the general will of his men, gave him some claim to call himself Imperator. (The sense of "one with the right to command" was probably lost, or at least not felt.)

It's important to emphasize a few things about this. First, all this happened under the Republic, and in no way made the victorious general at all monarchical or imperial (in our sense). Second, this was an honor bestowed by the soldiers, not by the Senate, and it was done by acclaiming him on the field of battle. This creates a distinction between what we might call capital-I Imperator (one who had been acclaimed by his troops after a victory) and small-I imperator (just a word for "general").

This all starts to come to a head with Julius Caesar. He was acclaimed multiple times, and not unreasonably felt that this mark of the common soldiery's affection gave him leverage against the Senate. Thus, when he staged his not-quite-revolution, no little of his appeal was that he was a man of the people and the soldiers, the sort of man who had a right to call himself Imperator. After Caesar's death, though, Augustus -- who was by no means as talented a general -- felt that Imperator was a grand title indeed, much too grand in fact to let just anyone enjoy, so he established it as a prerogative of the monarch, such that imperator began its evolution to "emperor."

Your final question, when people started considering themselves part of the Imperium Romanum, is related but distinct. In fact, using the word imperium to mean "empire," i.e. to refer to the total territory subject to Rome, dates to the Republic; the evolution of the term seems to be "the right to command" to "the areas where the right to command may be exercised" (i.e outside the city, on military campaign) to "areas conquered by someone with the right to command." In both the Republic and Empire, however, res publica was the preferred way to speak of the state; one would only call it imperium to emphasize its military aspects, especially its authority over subject peoples. This changed only very slowly, and it wasn't until much later that imperium became the normal word to refer to the Empire -- but even at the very end of the Empire in the West, authors could still refer to it as res publica. Perhaps people on the peripheries of the empire felt more strongly about it being an imperium than a res publica.

Edit: Related to Augustus' limit of the title imperator to himself and his successors, he also began limiting formal imperium, such that most generals were only thought of as exercising the imperium of the Princeps in absentia. This became important when a son of Crassus, one of the triumvirs, defeated an enemy army and personally killed the enemy king with his own hand in single combat. Such a deed had happened only three times before in Roman history, once by Romulus, and gave the doer the right to confer the spolia opima, an enormous honor. It was decided, however (in a clever bit of pseudo-archeology) that one had to exercise imperium on one's own behalf to earn the spolia opima, and thus that young Crassus didn't qualify. So a potential threat to Augustus (someone having more honor than him) was minimized.