The optimization process and recommendations:

1. 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:

CREATE INDEX cars_idx_sales_state_price ON "cars" ("sales_state","price");

The optimized query:

SELECT
        "cars"."id" 
    FROM
        de."cars" 
    WHERE
        "cars"."sales_state" = 'onsale' 
        AND (
            cars.is_disabled IS NOT TRUE
        ) 
        AND (
            cars.price >= 35920
        ) 
        AND (
            cars.price <= 659880
        ) 
        AND (
            "cars"."featuring_score" IS NOT NULL
        ) 
    ORDER BY
        CASE 
            WHEN cars.featuring_score < 'C' THEN 1 
            WHEN cars.featuring_score = 'C' THEN 2 
            WHEN cars.featuring_score > 'C' THEN 3 
            ELSE 4 END,
CASE 
    WHEN cars.au_rating >= 3 THEN 1 
    WHEN cars.au_rating = 0 THEN 2 
    WHEN cars.au_rating = 2 THEN 3 
    WHEN cars.au_rating = 1 THEN 4 
    ELSE 6 END,
CASE cars.brand 
    WHEN 'Audi' THEN 1 
    WHEN 'Alpina' THEN 2 
    WHEN 'Artega' THEN 3 
    WHEN 'BMW' THEN 4 
    WHEN 'Maybach' THEN 5 
    ELSE 6 END ASC,
CASE 
    WHEN ABS(cars.price - 347900) < cars.price * 0.2 THEN 1 
    WHEN ABS(cars.price - 347900) < cars.price * 0.4 THEN 2 
    WHEN ABS(cars.price - 347900) < cars.price * 0.6 THEN 3 
    ELSE 4 END,
CASE 
    WHEN de."cars".images_count = 0 
    OR de."cars".images_count IS NULL THEN 1 
    ELSE 0 END,
ABS(cars.price - 347900) LIMIT 61