This probably isn't terribly helpful, but when I lived in Iowa City, I looked for apartments in the
Daily Iowan classified section.
I graduated a while ago, so I'm not sure if things have changed, but I had a terrible time finding places to rent that would even allow cats, so good luck finding a place that allows a dog. I rented an apartment managed by the Chambers Agency for over a year that didn't allow cats and had them anyway, and never got caught, but that's risky of course.
If you have any other questions about Iowa City, let me know. It's a great little town.
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