You don't need a car in Berlin unless you need to move furniture or need to re-stock multiple beer crates at once.
car2go works fine for occasional small size shuttle needs (yet their coverage is shrinking with every update of their TOS). For the ultimate exposure to Berlin-style directness (some people mislabel this as rudeness) I would also recommend renting a Sprinter van at Robben-Wientjes.
As a rule of thumb:
if you stay within the S-Bahn Ring or in walking distance to any S/U-Bahn Station up to 3 stops out of it you get almost anywhere within the city in about half an hour. Biking is normally easier in the former east, since West-Berlin's post-war urban development was also all about cars.
Was in Texas multiple times for a few weeks and the car-centric AC-enabled lifestyle is always quite the culture shock for me.
It's spread out, but the S- and U-Bahns will take you quite far, out past the edges of the city. It's hard for me to imagine needing a car (OK multi-crate Bier runs maybe an exception, as another has written, or moving apartments). For real long-distance stuff there's the Regional Express and normal long-haul train service. As to walkability, I'm a dedicated walker and the city's fantastic for it. Sidewalks are quite wide (by US standards) and there are usually plenty of safe places to cross streets, legally and otherwise. Mind the Ampelmännchen and you probably won't even get hit by a trolley.
> OK multi-crate Bier runs maybe an exception, as another has written
You can generally have drinks delivered for little to no surcharge and probably cheaper than specifically renting a car. Plus you don’t have to carry them upstairs :)