Will a non-conviction appear on my live scan

Generally not. If you were taken to the drunk tank and later released without facing any charges, then the arresting/detaining police department should have given you a letter declaring your arrest to be a detention only. It should not show up on a background check. Have you considered running a background check on yourself?

It very likely will. Best thing to do now is to request your own livescan report and/or contact the California Dept of Justice and request your criminal history report.

I agree with Mr. Finnecy. It will likely show as an arrest. A petition for factual innocence would be your remedy for this, but it is unclear whether this is available to you. Speak with an attorney.