The fields that can be edited and displayed in a My Site's user profile page are controlled in the User Properties settings of your User Profile Service Application. In the case of your mobile number for example, you can set whether or not a user can modify the value in this property in the "Edit Settings" section of the individual property you are trying to control. Similarly, you can specify if the field should be displayed and where in the My Site profile page.
To answer your second question, a user will have left navigation links to "Apps, Documents and People" if he/she actually has a My Site/Personal Site. You need to determine if users can have a My Site in order to control these links. If you don't want users to have these links then you need to delete their My Site(s) and then set a policy that determines which users are (if any) allowed to "Create [a] Personal Site" in the "Manage User Permissions" of your User Profile Service Application.