что-то is when neither you nor your interlocutor know what this is
он что-то сказал - you both heard the sound but can't tell what he actually said
кое-что is 1. when you know what this is but your interlocutor doesn't, or 2. you both know what this is and by using this word you point for the other party at the information you share, kind of like a code
он кое-что сказал - he said something you should know or should've paid attention to or i won't tell you what it was (it's the connotation 1)
if we translate что-то as mashehu, кое-что i'd translate as dvar ma
нечто can be synonymous to что-то as well as to кое-что, but it differs by placement within sentence, it's rarely used in colloquial speech and usually accompanies adjectives: нечто большое, нечто синее... and adjectives in comparative: нечто более крупное, нечто покрупнее
что-нибудь/что-либо is anything in negative and interrogative sentences
in positive statements it means something in future tense
i will figure something out - я что-нибудь придумаю
in colloquial speech people frequently replace кое-что by что-то, however lexically it's wrong