The optimization process and recommendations:

1. Avoid Selecting Unnecessary Columns (query line: 2): Avoid selecting all columns with the '*' wildcard, unless you intend to use them all. Selecting redundant columns may result in unnecessary performance degradation.
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 `all_sent_Orders` ADD INDEX `all_orders_idx_nachforschungcompletedon` (`nachforschungCompletedOn`);

The optimized query:

SELECT
        * 
    FROM
        all_sent_Orders a 
    WHERE
        a.nachforschungCompletedOn IS NULL 
        AND CASE 
            WHEN a.lastMessageSent IS NOT NULL 
            AND a.lastResponse IS NULL THEN date(date_sub(current_date, INTERVAL 7 DAY)) = date(a.lastMessageSent) 
            ELSE (
                date(date_sub(current_date, INTERVAL 30 DAY)) = date(IFNULL(a.lastResponse, a.nachforschungSentOn)) 
                OR date(date_sub(current_date, INTERVAL 60 DAY)) = date(IFNULL(a.lastResponse, a.nachforschungSentOn)) 
                OR date(date_sub(current_date, INTERVAL 90 DAY)) = date(IFNULL(a.lastResponse, a.nachforschungSentOn)) 
                OR date(date_sub(current_date, INTERVAL 120 DAY)) = date(IFNULL(a.lastResponse, a.nachforschungSentOn))
            ) END