Russian has a more informal greeting «Приве́т» and a more formal «Здра́вствуй(те)». Here, we focus on the first, since it is the shorter one.
When on phone, use «Алло́» (Алё).
«Пожа́луйста» (please) has another popular position in the sentence—namely, after the verb (more on that later).
• you can also use «пожа́луйста» as a reply to "thanks", meaning "You are welcome!"
The phrase for "How are you?" literally means "How are your affairs (the stuff you do)?"
No one uses it as a greeting, i.e. you are not expected to use it with people you barely know (or those you know, for that matter). And be prepared for a person to actually tell you how they've been doing. ;)
Morning typically starts at 4 or 5 a.m., afternoon at noon, evening at 5 p.m. (at 6 for some) and night at 11 or at midnight.
You only use "Good night" (Споко́йной но́чи) when parting before sleep (or saying your goodbyes really late, so it is implied you or the listener are going to bed soon after).
If you are advanced enough to have noticed oblique forms used in some phrases—you are right! Greetings and other similar expressions are often shortened versions of longer phrases, where words still retain their forms. For example, «Споко́йной но́чи» probably replaces the longer «Я жела́ю вам споко́йной но́чи!» (I wish you a peaceful night). Needless to say, the full version is never used.