Here are a few choices....I know most apartments will take cats....dogs much more iffy.. You didnt say your price range...most of the more upscale apartment complexes will have 1BR 1BA for the 700-750 ish range.... Also how close to UND....walking distance or short drive? But I will give you links to three of the more recommended ones....
Newest
West Ridge Apartments and Legacy Apartments..these would be among the most upscale in town and do have underground parking included in each apartment...very nice to have. West Ridge is a bit closer to UND on S 42nd Street....but still 1-2 miles south of campus. Legacy is a bit farther south and east more toward 32nd Ave and its shopping.
West Ridge Apartments Grand Forks Area
Legacy Apartments : IMM Apartments: Over 5,000 apartments for rent
Gallery Apartments is known for its indoor pool and is very very close to UND....across the street in fact from far northwest end of campus living. It is not as new....but still nice.
The Gallery Apartments Grand Forks Area
There are many more to choose from but these stick out. There are new apartments (McEnroe apartments) being built off S 42nd street and east of the Alerus but it seems on the web site these are more 2-4 BR. And there is a Campus Place right across UND on 42nd St. Both of these are managed by Dakota Commercial.
Dakota Commercial
proptype=com_apartment
Do a search for IMM Apartments, Goldmark and Alerus Property Management for three of the bigger companies in town which own buildings....
I do want to point out the above apartments I mentioned I dont know for a fact they take cats....so please call them.
Good luck and glad to see your not afraid of the winter
Dan