Is shaking hands with one's wife upon entering the home from the authentic Sunnah, and does the Hadith: "Any man who enters his home and greets his family with peace and shakes hands with his wife and gently pats her hand, then their sins fall away or are shed just as the leaves of a tree shed in a wintry day," have a basis and is it authentic?

There is no reported hadith specifically about a man shaking hands with his wife upon entering the house. The hadith narrated from Abu Sa'id al-Khudri, "When a man looks at his wife... their sins fall from between their fingers," is weak. However, there is no religious impediment for a man to shake hands with his wife; rather, it is a completion of the greeting, and the husband will be rewarded for it if he intends a good intention thereby. There are general hadiths about the virtue of shaking hands and the virtue of giving salutations upon entering homes.