What is the wisdom behind a woman covering her face (wearing the khimar) when observing the 'iddah after her husband's death, and is it part of the Sunnah?

It is not sunnah (definition) for a woman in her 'iddah (definition) (waiting period) to always cover her face with the niqab, nor is it obligatory for her to do so. This is evidenced by the Prophet, peace and blessings be upon him, entering upon Umm Salamah and seeing her eyes while she was in her 'iddah. Had her face been covered, it would not have been possible to see them. The Prophet's silence regarding this is proof of its permissibility. The basic principle is that a woman in her 'iddah is like other women; it is not obligatory for her to cover her face except when seen by a non-mahram (definition) man.