That happens all the time, when people confess to a charge ahead of time, instead of proceeding to a trial. Remember that the purpose of the trial is to find out whether they are guilty when there is a factual dispute about that question. Here, I suppose the existence of a factual dispute is itself disputed: does that need to go to a jury, or is it enough that the trial judge and the appeal court looked at the record and decided there wasn't a dispute?
Confessing under the law is the same as being convicted though.
Certainly not always. Sometimes a person will confess a crime under immunity, and not be charged. Ulbricht didn't confess the murder for hire formally, and he wasn't charged with it. The controversy is that his role in it was used to influence the sentencing decision for a different crime.