MySQL - Multi level category structure

Given the following data and table structure:

id       name          parent_id
================================
1        food          NULL
2        snacks        1
3        cakes         2
4        birthdays     3

I would like to output these rows alongside all of their parent categories. So for example I would like to generate a query to output the data as follows:

id       name          parent_id    parent_1    parent_2    parent_3
===================================================================
1        food          NULL         NULL        NULL        NULL
2        snacks        1            1           NULL        NULL
3        cakes         2            1           2           NULL
4        birthdays     3            1           2           3

With this, I can easily get the IDs of every parent level of a given category.

I have tried doing this using sub queries but not quite managed to get it right. This is what I have tried:

SELECT id, name, parent_id, parent_id AS _parent_1,
(SELECT parent_id FROM categories WHERE id = _parent_1) AS _parent_2,
(SELECT parent_id FROM categories WHERE id = _parent_2) AS _parent_3
FROM `categories`

EDIT: Based on the feedback, it seems it's going to be somewhat difficult to get the data in the desired format.

Would it at the very least be possible to get all child categories of a given category? So for example for category ID 1, it should output the three categories below it (or four if it will include the given category itself).

You can simply self-join the table a number of times as required to meet maximum nesting depth. This could look like this:

SELECT
  c.id AS id,
  c.name AS name,
  parent.id AS parent,
  grandparent.id AS grandparent,
  greatgrandparent.id AS greatgrandparent
/* add other columns based on max depth of nesting */
FROM categories AS c
LEFT JOIN categories AS parent
  ON c.parent_id = parent.id
LEFT JOIN categories AS grandparent
  ON parent.parent_id = grandparent.id
LEFT JOIN categories AS greatgrandparent
  ON grandparent.parent_id = greatgrandparent.id
/* keep joining based on max depth of nesting */