Does the join order matter in SQL?

For INNER joins, no, the order doesn't matter. The queries will return same results, as long as you change your selects from SELECT * to SELECT a.*, b.*, c.*.

For (LEFT, RIGHT or FULL) OUTER joins, yes, the order matters - and (updated) things are much more complicated.

First, outer joins are not commutative, so a LEFT JOIN b is not the same as b LEFT JOIN a

Outer joins are not associative either, so in your examples which involve both (commutativity and associativity) properties:

a LEFT JOIN b 
    ON b.ab_id = a.ab_id
  LEFT JOIN c
    ON c.ac_id = a.ac_id

is equivalent to:

a LEFT JOIN c 
    ON c.ac_id = a.ac_id
  LEFT JOIN b
    ON b.ab_id = a.ab_id

but:

a LEFT JOIN b 
    ON  b.ab_id = a.ab_id
  LEFT JOIN c
    ON  c.ac_id = a.ac_id
    AND c.bc_id = b.bc_id

is not equivalent to:

a LEFT JOIN c 
    ON  c.ac_id = a.ac_id
  LEFT JOIN b
    ON  b.ab_id = a.ab_id
    AND b.bc_id = c.bc_id

Another (hopefully simpler) associativity example. Think of this as (a LEFT JOIN b) LEFT JOIN c:

a LEFT JOIN b 
    ON b.ab_id = a.ab_id          -- AB condition
 LEFT JOIN c
    ON c.bc_id = b.bc_id          -- BC condition

This is equivalent to a LEFT JOIN (b LEFT JOIN c):

a LEFT JOIN  
    b LEFT JOIN c
        ON c.bc_id = b.bc_id          -- BC condition
    ON b.ab_id = a.ab_id          -- AB condition

only because we have "nice" ON conditions. Both ON b.ab_id = a.ab_id and c.bc_id = b.bc_id are equality checks and do not involve NULL comparisons.

You can even have conditions with other operators or more complex ones like: ON a.x <= b.x or ON a.x = 7 or ON a.x LIKE b.x or ON (a.x, a.y) = (b.x, b.y) and the two queries would still be equivalent.

If however, any of these involved IS NULL or a function that is related to nulls like COALESCE(), for example if the condition was b.ab_id IS NULL, then the two queries would not be equivalent.

for regular Joins, it doesn't. TableA join TableB will produce the same execution plan as TableB join TableA (so your C and D examples would be the same)

for left and right joins it does. TableA left Join TableB is different than TableB left Join TableA, BUT its the same than TableB right Join TableA

If you try joining C on a field from B before joining B, i.e.:

SELECT A.x, 
       A.y, 
       A.z 
FROM A 
   INNER JOIN C
       on B.x = C.x
   INNER JOIN B
       on A.x = B.x

your query will fail, so in this case the order matters.