Delete parent if it's not referenced by any other child

Question 1

In PostgreSQL 9.1 or later you can do this with a single statement using a data-modifying CTE. This is generally less error prone. It minimizes the time frame between the two DELETEs in which a race conditions could lead to surprising results with concurrent operations:

WITH del_child AS (
    DELETE FROM child
    WHERE  child_id = 1
    RETURNING parent_id, child_id
    )
DELETE FROM parent p
USING  del_child x
WHERE  p.parent_id = x.parent_id
AND    NOT EXISTS (
   SELECT 1
   FROM   child c
   WHERE  c.parent_id = x.parent_id
   AND    c.child_id <> x.child_id   -- !
   );

db<>fiddle here
_{Old sqlfiddle}

The child is deleted in any case. I quote the manual:

Data-modifying statements in WITH are executed exactly once, and always to completion, independently of whether the primary query reads all (or indeed any) of their output. Notice that this is different from the rule for SELECT in WITH: as stated in the previous section, execution of a SELECT is carried only as far as the primary query demands its output.

The parent is only deleted if it has no other children.
Note the last condition. Contrary to what one might expect, this is necessary, since:

The sub-statements in WITH are executed concurrently with each other and with the main query. Therefore, when using data-modifying statements in WITH, the order in which the specified updates actually happen is unpredictable. All the statements are executed with the same snapshot (see Chapter 13), so they cannot "see" each others' effects on the target tables.

Bold emphasis mine.
I used the column name parent_id in place of the non-descriptive id.

Eliminate race condition

To eliminate possible race conditions I mentioned above completely, lock the parent row first. Of course, all similar operations must follow the same procedure to make it work.

WITH lock_parent AS (
   SELECT p.parent_id, c.child_id
   FROM   child  c
   JOIN   parent p ON p.parent_id = c.parent_id
   WHERE  c.child_id = 12              -- provide child_id here once
   FOR    NO KEY UPDATE                -- locks parent row.
   )
 , del_child AS (
   DELETE FROM child c
   USING  lock_parent l
   WHERE  c.child_id = l.child_id
   )
DELETE FROM parent p
USING  lock_parent l
WHERE  p.parent_id = l.parent_id
AND    NOT EXISTS (
   SELECT 1
   FROM   child c
   WHERE  c.parent_id = l.parent_id
   AND    c.child_id <> l.child_id   -- !
   );

This way only one transaction at a time can lock the same parent. So it cannot happen that multiple transactions delete children of the same parent, still see other children and spare the parent, while all of the children are gone afterwards. (Updates on non-key columns are still allowed with FOR NO KEY UPDATE.)

If such cases never occur or you can live with it (hardly ever) happening - the first query is cheaper. Else, this is the secure path.

FOR NO KEY UPDATE was introduced with Postgres 9.4. Details in the manual. In older versions use the stronger lock FOR UPDATE instead.

Question 2

delete from child
where parent_id = 1

After deleted in the child do it in the parent:

delete from parent
where
    id = 1
    and not exists (
        select 1 from child where parent_id = 1
    )

The not exists condition will make sure it will only be deleted if it does not exist in the child. You can wrap both delete commands in a transaction:

begin;
first_delete;
second_delete;
commit;