I believe one would use "хорошо" for "good" in that manner, though I am not a native speaker. My Russian teacher would use "хорошо" as a compliment when we did something well. добрый I think translates more directly as "kind" or "nice". Other than these greetings, it is used to describe people ("He's a good/nice guy"). Maybe a native speaker on here can help clarify?
Russian dictionaries present adjectives in their masculine form, so you would see "добрый" listed in bold.
утро: From Proto-Slavic *(j)utro (“morning, dawn”). From Proto-Balto-Slavic *auš(t)ra- (“dawn, morning”), from Proto-Indo-European *h₂ews-ro- (“of the dawn or morning, matutinal; eastern”), from *h₂ews- (“dawn; east”). Baltic cognates include Lithuanian aušrà, dial. auštrà (“dawn”), Latvian àustra, aũstra (“dawn”). Indo-European cognates include Ancient Greek αὔρᾱ (aúrā, “(esp. cool) breeze, fresh air of the morning”), Latin auster (“south wind”), Proto-Germanic *austrą (“east”), Proto-Germanic *Austrǭ (“Easter, springtime; name of a goddess” hence "East") (presumably from the goddess of the dawn, lust, fertility and spring, associated with the beginning of the year).
A variant *jutro appears in West Slavic, Slovenian and Serbo-Croatian, and is also attested in Old Church Slavonic, but considered secondary; there are further Slavic lexemes that display an alternation *ju- in West and South Slavic vs. *u- in East Slavic. Variants with *(j)ustr- point to *ustr- < *usr-, the loss of -s- being either due to dissimilation or perhaps due to the law of open syllables, if the -s- was variably assigned to the end of the first syllable. According to Kortlandt, the acute on the root implies a zero-grade variant (which would have parallels in Sanskrit उस्र (usrá-, “reddish, ruddy, bright, matutinal”), उस्रा (usrā́-, “dawn, morning”)), which, however, is not attested. The intrusive -t- is regular in Proto-Slavic, compare *ostrъ, *strumy.