Is it obligatory to make up the Asr prayer if the prostration (sajdah) was not completed fully due to a pebble, and that rakah was canceled, the prayer was completed, and the prostration of forgetfulness (sujood as-sahw) was performed after the salutation?

What the questioner did is incorrect, as it is obligatory to prostrate on the ground with part of the forehead, and it is not a condition to place the entire forehead on the ground. Her prostration is valid if part of her forehead touched the ground. As for her adding an extra unit (rak'ah) to the prayer based on an interpretation (ta'weel), this does not invalidate the prayer according to some scholars, so she does not need to repeat it.