Is it Sunnah to be known by a kunya such as "Abu Ahmad" or "Abu Sa'id," and is there a text that encourages this?

There is no harm in using permissible kunyas (teknonyms). It is an Islamic etiquette that Islam has encouraged. The Prophet, peace and blessings be upon him, used kunyas and gave kunyas to some of his Companions. The righteous predecessors also did the same.