I'm not exactly sure if they have to run a server. I wonder if there would be a way to know whether or not if they run one besides having the up-to-date blockchain but that doesn't mean their server would have the required power to produce a block.
To answer your second question, I think it's a question of efficiency and might also be tied to the fact that backup BP might not have to run a server.
I think Steem would benefit from having the witness be paid proportional to their amount of votes, not just so for the backup witnesses as it is now. This would create more competition among the top 20.