Is performing the funeral prayer for one's mother with bowing (rukūʿ) and prostration (sujūd) a sin, and what should be done to expiate for that, and is that equivalent to not praying over her at all?

The funeral prayer consists of four takbirat. After that, one recites Al-Fatihah, sends blessings upon the Prophet (peace be upon him), supplicates for the deceased, and then performs the taslim. There is no bowing (ruku') or prostration (sujud) in it. It is permissible to pray over a grave, even if a long time has passed. The Prophet (peace be upon him) prayed over a grave of someone buried at night and over a new grave when he was unable to pray at the time of burial. A number of Companions and Tabi'in prayed over graves after burial, including Aisha and Ibn Omar. Scholars have differed on the duration during which it is permissible to pray over a grave, with some holding that it is permissible indefinitely. Since you were an adult at the time of your mother's passing, it is permissible for you to pray over her grave now by positioning the grave in front of you, between you and the Qiblah. You should frequently supplicate and seek forgiveness for your mother and give charity on her behalf.