SQL delivery query performance
Distinct

In a properly designed database, this DISTINCT should not be needed:

  (SELECT COUNT(DISTINCT `ORDER_ID`)
   FROM `ORDERS`


If multiple different orders can have the same ORDER_ID, we're gonna have a bad time. 

Aliases

c, d, r... these are bad table aliases. Why? Because Mr. Maintainer has to go through your entire query to know what those mean, as they are not self-evident. Your table names are pretty short, but if you really need to shorten them further... perhaps something like crier, deliv, coord would be better. 

Join syntax

This USING command:

FROM `COURIERS` AS c
LEFT JOIN `DELIVERIES` AS d USING (`COURIER_ID`)


USING is not standard SQL syntax, when it comes to joins. Sure, it can be a nice shorthand in MySQL, but generally it is preferred to use standard SQL syntax (usually more verbose) instead of depending on syntactic sugar provided by a particular build or version of a language. Like this:

FROM `COURIERS` AS c
LEFT JOIN `DELIVERIES` AS d 
ON c.`COURIER_ID` = d.`COURIER_ID`


Performance

It's a bit difficult to assess performance hiccups if not presented with the execution plan, or a sample dataset to reproduce the issue. If I had to guess, I would say you have too many subqueries in your CASE statements. 

CASE statements are great and very fast when you use constants to compare; but when you use variable values, such as what you are doing... well... it's pretty slow. You're asking the SQL engine to run each subquery for each record essentially. It may find a way to optimize that's halfway efficient, but I believe you would be better using a set-based approach. Example: 

WHERE (CASE WHEN
         (SELECT MAX(`START_DATE`)
          FROM `DELIVERIES`
          WHERE `COURIER_ID` = c.`COURIER_ID`) IS NULL THEN d.`START_DATE` IS NULL ELSE -- etc.


Instead:

-- this goes before your CASE statement
INSERT INTO TEMP TABLE DeliveryDates
    SELECT IFNULL( 
        MAX( d.`START_DATE`, NULL ) 
    )
WHERE d.`COURIER_ID` = c.`COURIER_ID`


Then:

WHERE (CASE WHEN
         (SELECT `START_DATE`
            FROM DeliveryDates) IS NULL THEN d.`START_DATE` IS NULL 
        ELSE d.`START_DATE`