I'm not sure if it's OK to say "Sounds good" instead of "It sounds good".
If it is so, when can I omit "it"? Can I also say "Will rain", instead of "It will rain"?
The word being omitted is not it, but that:
"Shall I pick you up at 5:30?"
"That sounds good." [or] "Sounds good."
You can always leave out that or *that's" in quick assents:
"You'll come by at 5:30?"
"That's right." [or] "Right."
When you are making any kind of value judgment about something said immediately prior, leaving out that or that's is acceptable in informal contexts, particularly in conversation.
He says he graduated from Harvard. [That] Sounds fishy to me.
When you are making an assertion such as It will rain, the question of leaving out the it doesn't arise, for two reasons:
So the question about leaving out it is based on a misunderstanding of what's going on. But leaving off initial pronouns is quite common generally. It's called conversational deletion and there are several questions about it on this site.
English is a non- null-subject- language. It naturally follows that it is not a pro-drop language where verbal inflections are indication enough to assume the subject with least requirement of filling the syntactic gap by a dummy pronoun. Only in imperative mood, we drop pronoun (you), but again commission it into service for emphasis.
So every sentence must have a subject— real or its place holder we know by the name pronoun.(Learned critics have used different pronouns.) However they all fulfill the syntactic function of subjects. I have a special fascination for IT as it is the only impersonal pronoun we have and has many a fond name attached to it.
It is true that in day to day conversation and colloquial expression we often omit the subjects. Infants do often drop subjects. Of course, it's okey to say, " Sounds good" in informal conversation, novels, direct speeches or real life situations.
"Will rain.", can never be written since English doesn't function in null anaphora and requires the syntactic gap to be filled by Weather IT. This is yet another name of IT.