From my amateur reading of Mesopotamian history I understand that each city was tied to its own god or was thought of as the home of a god (I might be butchering that concept, but I think I'm reasonably close). If that's the case, did religious beliefs or teachings change as city-states rose and fell or power waxed and waned? If so, how did this happen?
You are correct in your understanding of Mesopotamian religion. The temple was understood quite literally to be the god's home on earth, and hence was a major economic powerhouse in society.
I will attempt to answer your question, but I am not an authority on mesopotamian religion, so it will probably be in more general terms. Hopefully someone else can come along and provide more clarity, and I will provide some books recommendations as well.
Mesopotamian religions was polytheistic, and like all religions that lasted over a long period of time, morphed. One major way this occurred was through contact with other religions, when gods would be syncretized into Mesopotamian religion. Another major development was the increasing prominence of the cult of Inanna/Ishtar. In relation to your specific question, as cities rose to power, they would have money to develop temples to their gods to show their devotion. In ANE devotion, the temple itself is the point, rather than our idea of inner devotion.
Books on mesopotamian religion:
Ancient near east