For cuttings, you wait until the cut edge dries, then you stick it in your succulent mix and they should root with no special care needed. I had these in landscape in SoCal and every now and then I would brush past them (they were easily 3' tall) and some branches would break off, fall on the ground, and root - with no care, no burying, and no rain.
Your mother tree doesn't need any special care - it should push new buds.
Don't forget - these are succulents and thrive on direct sun and heat. Do not allow them to stay wet. Keep them in full sun always. Water only when the soil is bone dry.