Chadron is a college town, so you're going to find most of the rentals are geared toward reducing the damage college kids could possibly do to the place.

Keep looking though. I lived in Chadron for five years, four in severa different rentals, and always with a dog. The trailer parks are strictly no dogs, now, though. (One of the them was in the process of implementing that rule when my then-boyfriend and I moved in with our two. We got grandfathered).
But there are still other options, some of which allow pets, especially if they can see that you're a grown-up.
Alliance is probably your best bet just because it's the biggest town. It's also a railroad town, so there are a lot of rental options because of it. So far as pets, no matter where you look, you'll just have to look harder for a place that allows them.
And there's nothing wrong with Rushville. Sheridan county seat, it's not the biggest town, but it's nice enough.

We lived south of Rushville for a couple of years.