reorder fields on user profile page
You need to go to "Admin > Configuration > People > Account Settings > Manage display". You will be able to reorder the fields from there. If you want to remove some of them from the display, you can select "Hidden" from the menu in the "Format" column.

EDIT:
The profile picture, however, is managed separately by Drupal, not as a field. If you want to be able to manage it, a workaround might be to disable the default Drupal profile picture (Admin > Configuration > People > Accounts > Personalization) and to create a new Image field to display the picture, which you can treat as any other regular field. There are even some modules that create the field for you, like user picture field.