I am someone who frequently has used the type of accommodation you could offer - staying in a room in someone's home while they live there. The audience is smaller, but it's there. And if your city is a big tourist destination, an affordable place to stay could be a big draw.
My recommendations:
- Keep the cats! You are right that it will be a plus for the right person. But make it really clear in the listing, and make sure to do very thorough cleaning in that room, perhaps keep them out prior to bookings?
- How comfortable is the sofa bed? I was going to say an actual bed is a must, but sofa beds have come a long way. I did stay on a friend's sofa bed that was more comfortable than most beds. The quality of the bed can impact the amount you charge.
- While most people visiting a city don't spend a lot of time in their accommodations, understand that you may have guests that want to prepare some meals there. Is that okay with you? In general, think about what beyond the room itself you are willing to consider common space.
- Offering coffee/tea is a very big plus. You could even set up a kettle, cups, and coffee/tea in the room if you want to encourage the idea that they should be in the room more than the kitchen

- Excellent wifi is a MUST for travelers
- putting together some sort of guide to your city, things to see, how to get around is a very nice touch. Some hosts have a transit card for guests to use, or some sort of pass - something that isn't a high cost to you but saves the guest the trouble of figuring it out, is really cool.
- I love when hosts have a guest book - prior guests write in the things they liked about the location, good restaurants nearby etc.
I just saw Louis' post, and while I will gladly own the "low-rent" traveler label, I strongly disagree that you need to price lower than a hostel. Shared-home Airbnb is what we do when we
don't want a hostel! Also, pricing it above hostels will keep out the folks who are doing things as cheaply as possible and don't really care where they are or how they impact their surroundings.
I think the biggest adjustment will be that you are going to have strangers in your home - what is the level of engagement that you are comfortable with? Yes, there will be annoyances like hair in the sink, possible cultural misunderstandings, a broken mug. But think about what you want out of this and create an environment that would be attractive to the market you want. And create a clear profile.
I suggest cruising around Airbnb and look for other shared-housing listings to see how others have positioned their rental for ideas. That should also give you an idea of pricing if you look in your city.
ETA: I know there's a Reddit forum for Airbnb hosts - can probably get some good information on the hosting side of things (both good and bad), if there is no one on FSU who's done that.