I want to create another input field like the categories. Whenever user posts a new question I want to have another drop down of locations associated with that particular question, like categories drop down in the the 'Submit Question' form. And how I can add a UI for managing it in the admin section like the one there exists for categories in Questions?
If you want to add extra fields to a post in the admin area, use the Advanced Custom Fields plugin. It is pretty good and has a lot of options.