Since there is possibility of "じゃありません" instead of "じゃないです" (which is logical, given "ありません" is polite "ない"), and it also seemed that I have seen "じゃ" used without "ない", I'd like to ask, what "じゃ" exactly means? It looks like some kind of particle which was used in old Japanese but is not used any more.
It's a contraction of
で, which has various uses, and the topic marker
は). It is not particularly related to classical Japanese, and is not used only with negation.