&& depends of the evaluation of the right expression when left one is true, || doesn't. You could rewrite it to:

if(
    $users == 'all' ||
    ($_POST['user'] == 1 && $users == 'admins') ||
    ($_POST['user'] == 0 && $users == 'mods')
)

And it'll be the same.

The operator precedence sheet you link to states as a separate note:

Although = has a lower precedence than most other operators, PHP will still allow expressions similar to the following: if (!$a = foo()), in which case the return value of foo() is put into $a.

So, in effect, an assignment inside an expression will be treated somewhat like a sub-expression. Exactly how and when this will happen isn't clear from the documentation, which just states that "similar" expressions will work this way.

The result is correct. PHP will perform the multiplication first and then the addition. It looks like this:

$res = 225 + (154 * 256) + (138 * 256 * 256) + (81 * 256 * 256 * 256)
$res = 225 + 39424       + 9043968           + 1358954496
$res = 1368038113

As your link https://www.php.net/manual/en/language.operators.precedence.php already explains, the table is ordered by precedence.

First, you've got in line 3 the type cast (https://www.php.net/manual/en/language.types.type-juggling.php#language.types.typecasting), than the 6th line (arithmetic operators).

Your code is equivalent to

Copy$x = ((float)'100') / 100;

or

Copy$y = (float)'100';
$x = $y / 100;

respectively.