PDO Exception Questions - How to Catch Them

You should look at the documentation. But If you dont find anything, you can add another catch :

<?php
try {
  $stmt = $db->prepare("INSERT INTO tbl_user (id, name, password, question, answer)    VALUES (NULL, :name, :password, :question, :answer)");
  $stmt->bindValue(":name", $_POST['name']);
  $stmt->bindValue(":password", $_POST['password']);
  $stmt->bindValue(":question", $_POST['question']);
  $stmt->bindValue(":answer", $_POST['answer']);
  $stmt->execute();
  echo "Successfully added the new user " . $_POST['name'];
} catch (PDOException $e) {
  echo "DataBase Error: The user could not be added.<br>".$e->getMessage();
} catch (Exception $e) {
  echo "General Error: The user could not be added.<br>".$e->getMessage();
}
?>

This must work because all exceptions of PHP plugins herits from the Exception native PHP class. (Since 5.0 if my memory is well).

PDO Exception Questions - How to Catch Them

As a rule -

DO NOT catch them.

For example, your code here should be written this way

$stmt = $db->prepare("INSERT INTO tbl_user (id, name, password, question, answer) VALUES (NULL, :name, :password, :question, :answer)");
$stmt->bindValue(":name", $_POST['name']);
$stmt->bindValue(":password", $_POST['password']);
$stmt->bindValue(":question", $_POST['question']);
$stmt->bindValue(":answer", $_POST['answer']);
$stmt->execute();
echo "Successfully added the new user " . $_POST['name'];

without any try or catch calls. Because you have no particular scenario for handling an exception here (a simple echo is scarcely counts as a handling scenario).

Instead, let it bubble up to the application-wide error handler (don't be scared by the term, PHP already has a built-in one).

However, I'm having issues catching the errors how I'd like (errors like "Duplicate Entry", "Null Value" etc in MySQL).

Only in case if you have a certain scenario, you have to use try-catch operator, but you have to always check, whether the error you've got is one you expected. Otherwise an exception have to be re-thrown:

try {
    $pdo->prepare("INSERT INTO users VALUES (NULL,?,?,?,?)")->execute($data);
} catch (PDOException $e) {
    if ($e->getCode() == 1062) {
        // Take some action if there is a key constraint violation, i.e. duplicate name
    } else {
        throw $e;
    }
}

and of course (as it turned out to be the vere problem for this question), you have to set up PDO in exception mode, either in a constructor parameter of simply by adding the code

$db->setAttribute( PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION );

right after connect.