What exactly does an employee back round check have in it for a Non-arrest (M) conviction soon to be dismissed have on it?

You're having a little problem with concepts and terms. The way you could get convicted for forgetting to show up in court is if the court made your lawyer go ahead with a trial without your presence. If you willfully fail to appear for your misdemeanor trial, a court can proceed without you.
Convictions don't usually get dismissed with the passage of time. Usually you have to go back to court to get an order formally dismissing the conviction. You can apply for a qualified dismissal under Penal Code sec. 1203.4. On a background check it will show that you were convicted but a court later dismissed the case under this section. It is a qualified dismissal because it does not erase the fact of the original conviction - only that after you were convicted a court dismissed it. You have to have satisfactorily completed probation and paid all the fines, fees, etc. in order to get relief under this Penal Code section.
Because you will be applying for licensure after your graduate, you will have to go through an application process again. With this in mind, gather all the court documents you can which reflect the facts of the conviction, ie. shooting gophers with a BB gun within the city limits. If your licensing agency asks for more information about this qualified dismissal, you will have all the paperwork ready to present.

If you have not been convicted of the charge it will not go on your record.

“Expungement” is a term used when referring to a process of cleaning up your criminal record. In this process you request that the court reopen your criminal case, withdraw the plea or guilty verdict, dismiss the charges, and re-close the case without a conviction. In effect, you are no longer a convicted person. However, the case record itself will still exist, but the outcome of the case in will no longer be your plea or
conviction, but instead dismissed in the “Interests of Justice,”
For most people, the benefits of an expungement far outweigh these restrictions. Under California Labor Code (Lab) § 432.7, employers cannot ask about arrests that did not end in conviction, convictions that have been judicially dismissed, or about any diversion or similar programs. If the employer is aware of any such arrests or programs, they cannot use this information to make hiring, promotion, or firing decisions.
Once a conviction is expunged, it becomes an arrest that did not end in conviction. Legally, you may answer “No” to these types of questions when applying with a private employer. Keep in mind, though, that background checks typically go back 10 years, and employers can see that you had a conviction dismissed. A better answer may be yes, but dismissed in the interests of justice. When applying for government employment, you must disclose the conviction and expungement. Talk to your attorney for further details.

You mentioned a lawyer in your post. Please contact that lawyer for direction.