As a previous travel nurse myself, I have rented entire houses (2BD, 1BA), 1BD condos, and apartments. It ranges based on the market and I mainly found rentals on VRBO and AirBnB. I never took housing from my staffing agency because most of the time they offer you something like a hotel room and I was always willing to dig for something better. As @Chris Allen mentioned, there are some great fb groups that are all about travel nurse housing - the one I look at is "Travel Nurse Housing - The Gypsy Nurse". It is literally a forum for people posting their places to rent and people asking for places to stay.
As to the second unit being unfurnished, I didn't know any travel nurses who traveled with furniture so renting stuff would be necessary. I literally drove across the country in my 2 door car and could only take what fit in my car.
Also, most travel assignments are 13 weeks - this varies, of course, but that is the standard. A lot of nurses will "extend" their contract and stay longer if they like the location or the hospital and can stay for almost a year before declaring residency. Just an FYI to keep in mind when booking.
I love this idea have definitely thought about implementing it as well! Good luck :)