is it illegal for the judicial system to convict someone without a trial?

What do you think a plea bargain is?   The person pleads guilty and he is considered convicted and gets sentenced. 

In the US it is, yes.

Douglas is an idiot.

No, there is no requirement that there be a trial.  

Pleaded guilty?   No reason for a trial. 

Everyone who gets a speeding ticket and just pays it is convicted without a trial.

If the defendant pleads guilty or no contest, that can forego any trial.

Charges of contempt can be asserted and ruled upon immediately without a trial.

No.  There can be no conviction without a trial in America, unless you're a fugitive from justice.