You're being very thoughtful about this, which is great!
First, I think you should consider (and you may have already) if you can fill these needs and wants with Uber and car sharing such as Zipcar. You could budget $150/month for those things and still spend less than you would if you bought a car. For reference, I have spend $217/month on expenses related to driving in the past 6 months. That is one $750 repair/maintenance bill plus insurance/gas/parking/registration. That is for a paid off 2003 Toyota Corolla.
If you decide you really want a car, given your use cases for the car, I think you would be best served by an older used car. Something in the 2008-2010 range. You could go a bit older, but that might mean more maintenance costs. I love hatchbacks in general and the Honda Fit specifically. I see a few on the Madison CL for $4000-$4500.
I would also probably be looking for a private sale, but in order to do that, you need a mechanic who you trust to look at the car before you buy it. I haven't done a private sale since the mid-90s, so someone else can talk about how that works better than I can.