DUI/DMV Records

A DUI conviction does not ever "disappear" from your record.  However, by virtue of something known as the "look back period", any prior DUI committed more than 10 years ago, is not considered when imposing a sentence for a subsequent DUI; the latter offense will be treated as your first since the court will only "look back" 10 years into your record.

This really has nothing to do with your situation however.  In your case you are asking will the fact that you did not complete your probation for a DUI conviction be an issue?  The answer is a definitive, "YES".  When you failed to complete the mandatory classes you violated your probation.  At that time a bench warrant was probably issued for your arrest.  Now with these type of warrants they authorities will not actively seek you out but if you are stopped by the police for something and they run a background check on you, this will all turn up.  You will then be arrested.

You need to consult with a DUI attorney as to all of this.  They will check on any outstanding warrants and will generally help you sort this all out.  Until you do speak with one, do not even think of applying for a license.  You could get arrested on the spot.