The optimization process and recommendations:

1. Avoid Subqueries (query line: 15): 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 `dept_emp` ADD INDEX `dept_emp_idx_dept_no_emp_no` (`dept_no`,`emp_no`);
ALTER TABLE `dept_manager` ADD INDEX `dept_manager_idx_emp_no_dept_no` (`emp_no`,`dept_no`);
ALTER TABLE `employees` ADD INDEX `employees_idx_emp_no` (`emp_no`);
ALTER TABLE `salaries` ADD INDEX `salaries_idx_emp_no` (`emp_no`);

The optimized query:

SELECT
        salaries.salary,
        employees.first_name,
        employees.last_name 
    FROM
        salaries,
        employees,
        dept_manager,
        dept_emp 
    WHERE
        employees.emp_no = dept_manager.emp_no 
        AND dept_manager.dept_no <> 'd002' 
        AND salaries.emp_no = dept_manager.emp_no 
        AND salaries.salary > ANY (
            SELECT
                salary 
            FROM
                salaries,
                dept_emp 
            WHERE
                dept_emp.dept_no = 'd002' 
                AND salaries.emp_no = dept_emp.emp_no 
        )