Most Koreans I've met translate 놀다 as 'to play'. But to address your question directly: 'to hang out' means to either linger in a location or accompany one or more people for a time with no specific purpose or agenda. So if you 'hang out at the mall' then you go to a shopping center without any specific intent to buy something. If you 'hang out with your friends' then you and your friends are together but are idle or at least don't have any specific plan about your activities.
Better remember it as a general location marker. At/in, depending on the context, both can be accepted.
If English accepts both equivalently, you should probably report it. They might just have forgotten some alternative answers.
However, if it the location is more specific, e.g. "the cat is in the box" as in "inside the box", you would have to use additional location information, e.g.상자 안에.
From what I understand, 에서 is used to indicate a place where an action is made, and 에 indicates an object is somewhere, as well as to indicate the time when something happens. There are exceptions though, like 가 (to go). With that verb you need to use 에 to say "go to" and 에서 to say "go from"