The optimization process and recommendations:

1. Avoid Subqueries (query line: 8): We advise against using subqueries as they are not optimized well by the optimizer. Therefore, it's recommended to join a newly created temporary table that holds the data, which also includes the relevant search index.
2. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `category` ADD INDEX `category_idx_name_lft` (`name`,`lft`);

The optimized query:

SELECT
        child.name,
        (COUNT(parent.name) - (parentDepth.depth + 1)) AS depth 
    FROM
        category AS child,
        category AS parent,
        category AS sub_parent,
        (SELECT
            child.name,
            (count(parent.name) - 1) AS depth 
        FROM
            category AS child,
            category AS parent 
        WHERE
            child.lft BETWEEN parent.lft AND parent.rgt 
            AND child.name = 'ELECTRONICS') AS parentDepth 
    WHERE
        child.lft BETWEEN parent.lft AND parent.rgt 
        AND child.lft BETWEEN sub_parent.lft AND sub_parent.rgt 
        AND sub_parent.name = parentDepth.name 
    GROUP BY
        child.name 
    HAVING
        depth > 0 
    ORDER BY
        child.lft