I believe, it should be rather a question of professionalism, than a question of popularity. We all know, how to measure popularity. But how to measure skill and understanding?
I would advice to have 2 chambers in that Council, one made of elected players, just like it is described in the topic-post. And the second made of owners of top servers, e.g. red shift, elwyn, etc, or their representatives, selected by owners in case they do not wish to bear these duties themselves.
This would make it possible for the community to be represented by both players and server-admins, mirroring two different game experiences - the gaming one and the server-maintaining one, which are totally different.
Also, this would actually help simple players to compete in voting vs our beloved admins, as the second ones have far more options in gaining election power.
Also, to the questions you have asked:
Complete number of councillors - about 20.
Requirements: no age, sex or location restrictions. The applicant should be either a well-known member of the community, either a community-builder, keeping a well-known server.
That is quite enough itself, though restricts many people by location, as I believe, no chinese or north korean player will be elected in any way, especially if he(she) doesn't speak English.
Field of authority of the council, duties and functions - we all are aware of how and WHY minecraft has died. In brief - the devs were too far from the community. So, that's the main purpose of any community manager (and the Council is just a fine name for a bunch of ones). Keeping the game alive by giving adequate feedback for the devs, making it possible for the game to develop properly. I must admit, it will be up to the devs to listen, as no advice can reach a deaf person.
What should the application questions be? Though I stated above that no significant restrictions should be made for applicants, I still believe their application CVs should be as complete, as possible, such info as ASL, known foreign languages, gaming experience, overall life experience, education, sci-fi reading experience and likes-dislikes, any notable achievements he(she) would like to mention, also downloadable models of ships\bases the applicant has built in SM, a short essay of the applicants' own view on the future of SM, including his own list of features\suggestions, that he(she) is eager to see in game in the future. I also would strongly advise including a short video self-presentation (not less than a minute) and making it obligatory for application.
Finally I would like to express my deepest condolences to that person from the Shine staff, who will be in charge of these elections, as they gonna be an enormous pain in the ass, full of dirt, intrigues, forum rage wars etc etc etc. Prepare to have much more white hair on your had, pal, especially if you won't be able to take it easy.