Mapping fields of weakly related tables in SQL

Question 1

Using the table definition from @SoulTrain (but requiring the data to be sorted in advance):

; WITH ORDERED_PAIRINGS AS (
    SELECT i.ITEMID, b.BUCKETID, ROW_NUMBER() OVER (ORDER BY i.SIZE, i.ITEMID, b.SIZE, b.BUCKETID) AS ORDERING, DENSE_RANK() OVER (ORDER BY b.SIZE, b.BUCKETID) AS BUCKET_ORDER, DENSE_RANK() OVER (PARTITION BY b.BUCKETID ORDER BY i.SIZE, i.ITEMID) AS ITEM_ORDER
    FROM @ITEM i
    JOIN @BUCKET b
      ON i.SIZE <= b.SIZE
), ITEM_PLACED AS (
    SELECT ITEMID, BUCKETID, ORDERING, BUCKET_ORDER, ITEM_ORDER, CAST(1 as int) AS SELECTION
    FROM ORDERED_PAIRINGS
    WHERE ORDERING = 1
    UNION ALL
    SELECT *
    FROM (
        SELECT op.ITEMID, op.BUCKETID, op.ORDERING, op.BUCKET_ORDER, op.ITEM_ORDER, CAST(ROW_NUMBER() OVER(ORDER BY op.BUCKET_ORDER) as int) as SELECTION
        FROM ORDERED_PAIRINGS op
        JOIN ITEM_PLACED ip
          ON op.ITEM_ORDER = ip.ITEM_ORDER + 1
         AND op.BUCKET_ORDER > ip.BUCKET_ORDER
    ) AS sq
    WHERE SELECTION = 1
)
SELECT *
FROM ITEM_PLACED

Question 2

Try this... I was able to implement this using recursive CTE, all in 1 single SQL statement

The only assumption I had was that the Bucket and Item data set are sorted.

DECLARE @BUCKET TABLE
    (
     BUCKETID INT
     , SIZE INT
    )

    DECLARE @ITEM TABLE
    (
     ITEMID INT
     , SIZE INT
    )
    ;  
    INSERT INTO @BUCKET
    SELECT 1,2 UNION ALL
    SELECT 2,2 UNION ALL
    SELECT 3,2 UNION ALL
    SELECT 4,4 UNION ALL
    SELECT 5,4 UNION ALL
    SELECT 6,7 UNION ALL
    SELECT 7,9 UNION ALL
    SELECT 8, 11 UNION ALL
    SELECT 9, 11 UNION ALL
    SELECT 10,12 

    INSERT INTO @ITEM
    SELECT 1,2 UNION ALL
    SELECT 2,2 UNION ALL
    SELECT 3,5 UNION ALL
    SELECT 4,11 UNION ALL
    SELECT 5,12;

    WITH TOTAL_BUCKETS
    AS (
        SELECT MAX(BUCKETID) CNT
        FROM @BUCKET
        ) -- TO GET THE TOTAL BUCKETS COUNT TO HALT THE RECURSION
        , CTE
    AS (
        --INVOCATION PART
        SELECT BUCKETID
            , (
                SELECT MIN(ITEMID)
                FROM @ITEM I2
                WHERE I2.SIZE <= (
                        SELECT SIZE
                        FROM @BUCKET
                        WHERE BUCKETID = (1)
                        )
                ) ITEMID --PICKS THE FIRST ITEM ID MATCH FOR THE BUCKET SIZE
            , BUCKETID + 1 NEXT_BUCKETID --INCREMENT FOR NEXT BUCKET ID 
            , (
                SELECT ISNULL(MIN(ITEMID), 0)
                FROM @ITEM I2
                WHERE I2.SIZE <= (
                        SELECT SIZE
                        FROM @BUCKET
                        WHERE BUCKETID = (1)
                        )
                ) --PICK FIRST ITEM ID MATCH
            + (
                CASE 
                    WHEN (
                            SELECT ISNULL(MIN(ITEMID), 0)
                            FROM @ITEM I3
                            WHERE I3.SIZE <= (
                                    SELECT SIZE
                                    FROM @BUCKET
                                    WHERE BUCKETID = (1)
                                    )
                            ) IS NOT NULL
                        THEN 1
                    ELSE 0
                    END
                ) NEXT_ITEMID --IF THE ITEM IS PLACED IN THE BUCKET THEN INCREMENTS THE FIRST ITEM ID
            , (
                SELECT SIZE
                FROM @BUCKET
                WHERE BUCKETID = (1 + 1)
                ) NEXT_BUCKET_SIZE --STATES THE NEXT BUCKET SIZE
        FROM @BUCKET B
        WHERE BUCKETID = 1

        UNION ALL

        --RECURSIVE PART
        SELECT NEXT_BUCKETID BUCKETID
            , (
                SELECT ITEMID
                FROM @ITEM I2
                WHERE I2.SIZE <= NEXT_BUCKET_SIZE
                    AND I2.ITEMID = NEXT_ITEMID
                ) ITEMID -- PICKS THE ITEM ID IF IT IS PLACED IN THE BUCKET
            , NEXT_BUCKETID + 1 NEXT_BUCKETID --INCREMENT FOR NEXT BUCKET ID 
            , NEXT_ITEMID + (
                CASE 
                    WHEN (
                            SELECT I3.ITEMID
                            FROM @ITEM I3
                            WHERE I3.SIZE <= NEXT_BUCKET_SIZE
                                AND I3.ITEMID = NEXT_ITEMID
                            ) IS NOT NULL
                        THEN 1
                    ELSE 0
                    END
                ) NEXT_ITEMID --IF THE ITEM IS PLACED IN THE BUCKET THEN INCREMENTS THE CURRENT ITEM ID
            , (
                SELECT SIZE
                FROM @BUCKET
                WHERE BUCKETID = (NEXT_BUCKETID + 1)
                ) NEXT_BUCKET_SIZE --STATES THE NEXT BUCKET SIZE
        FROM CTE
        WHERE NEXT_BUCKETID <= (
                SELECT CNT
                FROM TOTAL_BUCKETS
                ) --HALTS THE RECURSION
        )
    SELECT 
        BUCKETID
        , ITEMID
    FROM CTE

Question 3

I'd say that a single SQL query is maybe not the tool for the job as the buckets are "consumed" by allocating items to them. You can use SQL, but not a single query. suggestion in pseudocode below:

Have a cursor on ITEM: 
    within the FETCH loop for that {
    SELECT in BUCKET the bucket with minimum bucket id and bucket size >= item size 
    INSERT bucket id, item id to MAPPING
}

If you need the NULL (unoccupied) buckets, you can locate them via a further 
INSERT into MAPPING (....)
SELECT <bucket id>, NULL
from    BUCKET 
where <bucket id> not in (SELECT <bucket id> from MAPPING);