Add additional setting per flag for roles that can change other peoples' flags

So I set the flag to "Users may only flag content they own". This works but there's a problem. No one, not even UID 1, is able to (un)flag content that they don't own.

That's right. Flag currently doesn't give user #1 preferential treatment.

I'd like to have a role that can be set on a flag where members of that role are allowed to alter other peoples' flags. If that's not possible, could it at least be done so UID1 can edit any flag?
[...]
I'm dealing with global flags

A quick answer:

You can implement hook_flag_accees() in your module to tweak Flag's access scheme. For example, to give user #1 permission to use a flag unconditionally, do:

/**
 * Implementation of hook_flag_access().
 */
function MYMODULE_flag_access($flag, $content_id, $action, $account) {
  if ($flag->name == 'needs_work' && $account->uid == 1) {
    return TRUE;
  }
}

/**
 * Implementation of hook_flag_access_multiple().
 *
 * You must also implement this hook. This hook, for performance reasons, is called for a bunch of items.
 */
function MYMODULE_flag_access_multiple($flag, $content_ids, $account) {
  $access = array();

  if ($flag->name == 'needs_work' && $account->uid == 1) {
    foreach ($content_ids as $id => $action) {
      $access[$id] = TRUE;
    }
  }

  return $access;
}

However, your module must come after Flag in the modules' chain (i.e., have greater weight or come later alphabetically) or else Flag's own permission scheme would override yours. Indeed, I see that others have already noticed this flaw in Flag:

#720672: Make $flag->access (and _multiple) act in a consistent manner

We should certainly improve/fix matters before Flag 2.0 is out. I'll be investigating this.

Add additional setting per flag for roles that can change other peoples' flags
[...]
I'm filing it as a feature request

(An alternative is to have a "super flagger" role: a user carrying it would bypass access checks. But this solution isn't granular enough. So yours is better.)

This sounds to me like a very valid feature request. On the other hand, we want to keep Flag small and simple, and that feature can be easily implemented by a contrib module (via hook_flag_access()).

========

In my searching, I found #755840: Administrator Role to Change other User's Flags or Flag on behalf another user It's not quite a duplicate, though, because I'm dealing with global flags

Indeed, #755840 is irrelevant for you.