"Ok. But how could the galaxy have such a well developed spiral shape? Or perhaps that is why it has a spiral shape - because the stars on the outer rim (like ours) haven't gone around as often as the stars nearer the center...."

This is a question that puzzled astronomers for many years and is still not entirely resolved. The reason it was such a puzzle is that looking at other galaxies at large distances (and thus at much younger ages) they have much the same distribution of spiral formations as nearby galaxies, including the Milky Way. This means that no matter how many times a galaxy has rotated, it's spiral structure (if it has one) does not seem to change. If it were the result of differential rotation the spiral arms would gradually stretch out farther and farther until they were no longer recognizable. Also it has been found by measuring the rotational velocities of galaxies, including the Milky Way, that there actually is very little variation in velocities for stars at different distances from the galaxies center, once you cross a certain distance from the center, and thus little differential rotation.
The current and fairly well supported theory to explain spiral structure is that it does not result primarily from differential rotation but rather as a result of a gravitational 'wave', analogous to ripples on a pond when a rock is thrown in. The big question now is what is driving the wave. The prime suspect is (not surprisingly) supermassive black holes at the center of galaxies, but not all aspects of the phenomina can be explained this way.