Gymkhana, thank you for your thoughtful questions.

As to your first, one would speak of "observing" this holiday.

Your second is more complicated. A standard jewish expression for all holidays (indeed, jocularly used among jews when non-jewish holidays provide a day off work) is good yontif, wishing a "good holiday". Many older jews use the pure yiddish form: gut yontif (first word rhyming with "foot"). The yiddish yontif, meaning holiday, comes from the hebrew yom (as in Yom Kippur) = day, and tov = good [masc. form]).

Shana tovah (= "good year") is the greeting specific to the high holidays, Rosh Hashannah and Yom Kippur, but is more applicable to Rosh Hashannah. Hebrew: shana = year; hence Rosh Hashannah (the jewish new year's day) = head of the year; and to say shana tovah is to wish one a good year.

Jews observant enough to fast on Yom Kippur might say gemar tov (good fast) or variants, but this is uncommon.

The most precious wish comes from the legend that the Lord chooses, each Yom Kippur, which persons shall be granted the blessing of another full year of life. To closest family members one might say, on Yom Kippur, "May you be inscribed in the Book of Life."