ね, the 已然形 (realis) form of the Classical Japanese auxiliary verb ず
ば, basically the same as modern ば
So to answer your questions in detail (great format, by the way!):
It is a different construction -- actually なければ is a reinvention of the same concept (已然形 + ば) using modern parts!
It is a very old form, here it is in the Manyoshu: "... 君が目見ねば苦しかりけり" ("... Not seeing you (lit. your eyes), I suffer." -- note the use of the 已然形 was a bit different back then. This changed a few hundred years ago (IIRC) to something more like our usage)
I don't think its usage is especially limited except insofar as it is a bit of an archaism. (Note that the first two examples include ならぬ and 倒る, also archaisms.) I think most people would recognize it, but few would use it except for intentional effect (intentionally overformal speech as joke, etc.). However, this kind of pseudo-archaic talk is quite popular among certain internet circles, so it might be more common and "normal" there, and might have seeped out into real life from that point. I hope that someone else will be able to provide better info on this point.
As above, it is a bit of an archaism and so while not informal as such, it might be considered inappropriate or unusual in a case where standard Japanese was expected.
なきゃ and なくちゃ are normal standard Japanese, just informal (as you know of course), so it is not like them.