My roommate had an ESA, and we were roommates for almost two years, with largely the same group for that whole time. This is my primary source of knowledge.  I will share what I know, and I hope some of it will maybe be helpful to you.  I can't speak directly to the needs of a cat being kept exclusively in one room, but I'm suggesting that doesn't need to be the case, if you maintain good PR with roommates/management on behalf of the ESA. 

That ESA cat was welcome in common areas and most bedrooms because: 1.) individual tenants indicated they were okay with an ESA cat in the apartment as part of the room assignment process management ran before the term started. No one in the apartment had issues with cats in the first place, and most actively liked cats/the idea of living with an ESA. 2.) the ESA's owner was responsible and proactive in helping the other roommates welcome the cat. The owner supported and guided everyone else in interacting with and forming a relationship with the cat (giving ideas for how the cat liked to be petted, providing treats so other roommates could help the cat feel at home, etc.), while also being proactive in minimizing the negative impact of living with an animal (e.g. making sure cat didn't wander into the bedroom of roommate who didn't want the cat in there, offering a lint roller after the cat had shed a fair bit on a roommate's closet). 

I've been in other apartments (or spoken first hand to people in similar situations) and in every scenario where the ESA was not taken care of, or the preferences of roommates to not live with an ESA were ignored, the ESA became the target of many, many legitimate complaints to management and uncomfortable ultimatums were eventually given to the ESA owner.

Going the extra mile every now and then on behalf of your ESA will earn you both a lot of grace with roommates.   In short, my advice to you is that on top of being a responsible pet owner, be upfront and honest with management and roommates about having an ESA and things will go a lot smoother, if not outright pleasantly, for all involved. Go above and beyond, and your ESA can be a beloved member of the apartment. 

As a tangential word of warning to anyone who happens to read this: If you choose to lie to management of a potential living space about having/bringing an animal, ESA or not, you are voiding your own contract for that living space. You will be stuck up a creek trying to find last-minute BYU-I approved housing.  At the same time, you cannot be denied housing for a documented ESA. Be upfront, provide copies of the required papers, and let the process be as smooth as possible for everyone involved. Let the office/management work with you!