As the comments show, recursive queries are not possible. So I fear, there will be no satisfying answer to your first question.
However in the second question you mention you want to have all superparent users, who have at least 2 children (and here I assume you mean at least two levels of children).
SELECT id, name, CONCAT(level,children1,children2) AS children
FROM
(SELECT
t1.id,
t1.name,
IF(t2.name IS NOT NULL, t1.level + 1, t1.level) AS level,
IF(t1.children1 IS NULL, '', CONCAT('(',t1.children1)) AS children1,
IF(t2.name IS NULL, IF(t1.children1 IS NULL, '', ')'), CONCAT(', ', t2.name, ')')) AS children2
FROM
(SELECT u1.id, u1.name, u2.id AS bridge, u2.name AS children1, IF(u2.name IS NOT NULL, 1, 0) AS level
FROM users u1
LEFT JOIN users u2 ON u1.parent = 0 AND u2.parent = u1.id) t1
LEFT JOIN users t2 ON t2.parent = t1.bridge) x
This will retrieve all superparents (parent=0) and their first two levels of children (Answer to question 1 with a defined level). If you add
WHERE level > 1
to the query, you will get the filtered list of all superparents who have at least 2 levels of children (Answer to question 2).
Getting all children to a superparent is actually only hard because in a row the parent value is saved but not the child's value (if in your schema elements do not have siblings). The other way around is fairly easy by using incremental variables. In your case, if you'd want to find the complete heritage of D, you could run
SELECT t2.id, t2.name, level
FROM (
SELECT @r AS _id, (
SELECT @r := parent
FROM users
WHERE id = _id
) AS parent, @l := @l + 1 AS level
FROM (
SELECT @r := 4, @l := 0
) vars, users u
WHERE @r <> 0
) t1
JOIN users t2 ON t1._id = t2.id
ORDER BY t1.level DESC
where @r is initially set to D's id value. The query returns the element itself and each parent in separate row along with the reversed level:
id name level
1 A 3
3 C 2
4 D 1
Of course, this can only be run separately for each element because of the dynamic variables, but it gives you the complete line from a child up to the uppermost parent.