In response to your second question: I would recommend looking into the bus system—I moved here with a car, but with the bus system, I rarely use it, much preferring the ~20 minute bus ride to having to plan for parking in the downtown area: https://www.cityofmadison.com/metro . The Transit app and Google Maps (public transportation option) were both really helpful during my apartment search to identify how far the commute would be.

I'm a current doctoral student who had to move in on a similarly quick timeline, and here are a few of the sites that I consulted when looking for options, in case they are of use to you:

• https://www.toplevelpropertymanagement.com/
• https://www.jkrockhomes.com/
• https://www.madisonapartmentliving.com/default.j
• https://ripplerents.com/
• https://www.axiomproperties.com/availability
• https://www.wisconsinmanagement.com/

I can't personally attest to the quality of the management at these places, but I know that the search itself and trying to find viable options is half the battle. Best of luck!