The runner at first had taken a lead off the bag. The liner to Short would have required the runner to tag-up to advance. Since the Short stop tagged the advance bag before the runner got back to first to tag up the base runner is out.
No the play is at the bag the runner left. You can tag the base ahead of the runner because there is no force play there.
Nope. Not the rule. You can’t tag the bag ahead of the runner in that case. You must either tag the runner or the base he failed to tag-up from.