What do I file after the death of my mom?

Based on your post, it appears that probate may not be necessary but, if there was a will, that will should still be placed on deposit with the clerk and the death certificate (without cause of death) should still be recorded in the public records of the county where decedent passed away. Also, it would still be prudent to consult with an attorney to ensure that indeed no court action must be taken. All the best.