2 Corinthians 13:5 - Test yourself, Paul says. Too often we are too busy testing others rather than ourselves (although in verse 6 he implies that they would evaluate Paul himself). But look what he says as well - "Or do you not realize this about yourselves, that Jesus Christ is in you?" I think he's reminding them again that even though their sin is resent in themselves as the examine, they must not forget that Christ is in them. But even more, as they see who they are, they are to remember to whom they belong. Is what they see worthy of what they have been given? So it's both a reminder to take heart because Christ in in them and to be sober, contrite and repentant, because Christ is in them.
2 Corinthians 13:6-7 - More important to Paul than if they believe that he has 'not failed the test' is that they would be righteous even if they believe he to have failed. He's more concerned about their hearts than his reputation. The implication is that he desires them to show mercy to the sinner instead of judgment, but perhaps he's felt more the opposite. And it's not the sting of the judgment that grieves him, it's that they are not yet what Christ would make them to be.
2 Corinthians 13:11 - Do we often 'aim for restoration'? Not nearly often enough. Rather, mostly we settle for peaceful division or at worst we target and fire in judgment.