You're thinking of Zipcar.

With Car2Go, you leave the car in any legal on-street parking space (sometimes limited to 15 min+/1 hr+/2 hr+ depending on city) in the "home area" (borders depend on the city but are typically pretty expansive). The company contracts with each city to basically pay a flat rate for street and metered parking for all the cars yearly.

No, you can leave it nearly anywhere in the service area. The particular rules are up to the city you're in; in Seattle they amount to "you can park anywhere, provided it allows a normal car to park for at least two hours, and doesn't become illegal to park during certain times (e.g. bus lanes)".

I gather from their website the car may be returned to "any non-restricted curb side parking throughout the car2go Home Area"

How extensive the home area is depends heavily on the city. In Los Angeles, it looks like a tiny sliver of the city [1] whereas in Berlin the coverage looks a lot better [2] (although it's possible I'm being confused by different map zoom levels)

[1] https://www.car2go.com/en/losangeles/ [2] https://www.car2go.com/en/berlin/

Car2go can be parked at any street legal parking spot with 2+ hr limits. There are no restrictions really.