There's a version that «пока» as a farewell phrase originated from something like «while we are away, be OK» or a phrase like that.
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The connection seems clear enough: https://en.wiktionary.org/wiki/%D0%BF%D0%BE%D0%BA%D0%B0#Adverb
It's loosely comparable to how English speakers may part with a very informal "Later," shortened from "See you later."
Yes. It's a complex sentence with two clauses. Each clause has a different subject-predicate pair (брат работает, я готовлю), so it's basically two sentences stacked into one. Russian almost always requires a comma between clauses (in rarer cases, a colon, semicolon, or a dash might be used). If a clause is embedded inside another clause, we would put commas before and after it.
Present (both verbs, «рабо́тает» and «гото́влю»).
The past tense would be «Мой бра́т рабо́тал, пока я гото́вила» (if «я» is a girl), or «Мой бра́т рабо́тал, пока́ я гото́вил» (if «я» is a boy).