The biggest problem that I see with a self-pardon is that Trump would essentially be serving as his own judge (nemo judex in re sua (roughly, no one is the judge in his own case)), which goes against principles of natural justice.
As for standing, I don't practice in the criminal arena, but I would think the Department of Justice would have standing to bring the case against Trump. The case, of course, would not be for the use of the pardon itself, but rather the offense for which Trump pardoned himself (Trump would then argue the pardon, and the counter would be that the pardon was unconstitutional/illegal/abuse of power/etc.). As to your second question, I believe Burdick would stand for the proposition that Trump has confessed the pardoned offense. As Handyman alluded to, Trump may be able to retroactively reject the pardon, and then make a coercion type argument (which would be hilarious, as it would essentially mean that Trump would be arguing that he coerced himself into accepting the pardon). Without having done the research, I'm not sure if courts have addressed that issue, but I can't imagine in this factual scenario it would be a winner.
