Any formula for final sentences in 長爪梵志請問經 (Dirghanakha-sutra)?
1) Yes, "皆大歡喜，信受奉行" is one of the common closing sayings in the sutras. For example,

皆大歡喜，信受奉行。《金剛般若波羅蜜經》
皆大歡喜，信受奉行。《大方廣圓覺修多羅了義經》
悉皆歡喜，信受奉行。《占察善惡業報經》
皆大歡憙，信受奉行，禮佛而退。《佛說月光菩薩經》

The other common closing saying is "皆大歡喜，作禮而去". For example,

諸比丘聞經歡喜，為佛作禮而去。《六度集經》
諸比丘聞佛所說，皆大歡喜，為佛作禮。《佛說菩薩本行經》
聞佛所說，歡喜信受，作禮而去。《佛說阿彌陀經》
一切大會，皆大歡喜，受持佛語，作禮而去。《妙法蓮華經》
皆大歡喜，作禮而去。《楞嚴經》
聞佛所說，皆大歡喜，禮佛而退。《佛說觀無量壽佛經》



2)
皆 all (the crowd)
大 very; greatly
歡喜 pleased; glad; joyful; happy; delighted; gleeful

皆大歡喜: "all the [sentient beings] rejoyced greatly"
Yes, the translation is correct.

信 to believe
受 to accept
奉 respectfully
行 to perform; to practice

信受奉行
believe, accept and respectfully perform/practice the teaching.

作禮 to salute
而去 then leave

作禮而去
salute and then leave