I have the answer for question 2, I didn't put the use_checkbox at the right place! 😓
array( 'type' => 'categories', 'label' => $this->l('Featured categories'), 'name' => 'FEATURED_CATEGORIES', 'tree' => array( 'id' => 'category', 'selected_categories' => array((int)Configuration::get('category')), 'use_checkbox' => true ) ),
But it still collapse by default and I still don't have any clues on how to save the selected_categories.