First, let me comment on your three examples:
です ⇔ であります
We discussed です before. According to 大辞林, there are several theories, but we don't know its etymology for sure. This is one of the three theories it lists, though. I've read that でございます may be more likely, but I never read an explanation why, so I won't make that assertion here.
じゃない ⇔ ではない
Keep in mind there are two distinct
じゃs. This one is a contraction of では. The other is historically spelled ぢゃ and contracts from である→であ, just like だ.
だろう ⇔ であろう etc.
Yes, that's right.
Is it accurate to call な an inflection of だ? Since である can also replace な anywhere (as far as I can tell), could it be possible that な and だ derive from the same word, which through sound change has come to be differentiated according to function (終止形 and 連用形)?
Historically, である derives from にてあり (see 大辞林 again), which is にて plus あり. The combination にて is in turn に plus て. In にてあり, the classical あり became ある, probably before にて became で, so the progression looks something like
d'a < de a < de aru < nite aru < nite ari.
The progression of な is pretty similar, but without the て. The combination に plus あり became なり, then なる (note: not the same as the verb なる "to become"), and then it lost its る, becoming simply な. (Note that this is a slightly different path than だ took, since there was never a だる.) So the big etymological difference between the two is, more or less, whether they contain a て, and our progression for な is
n'a < n'aru < n'ari < ni ari.
Finally, で is historically the particle で, which again derives from にて. The difference in this case is that the ある (historically あり) is missing, because the sentence is still going. (Something like A にて B にてあり, I think.) We get the simple progression
de < nite.
So yes, they're closely related. And yes, I think most grammars consider them inflections of the same thing.
Remember, though, that diachronic and synchronic analyses are two different things, and one doesn't necessarily determine the other. The above is not how they're normally analyzed in modern Japanese, in which they're all reanalyzed as forms of だ. To be specific, I believe で is reanalyzed as a 連用形 form of だ; the combination である is reanalyzed as our reanalyzed で (which is 「だ」の連用形) plus ある; and な is likewise analyzed as the 連体形 of だ.
But then, Japanese school grammar doesn't use the term "copula" at all (I think the closest it gets is something like 「断定の助動詞」). Perhaps ironically for this question, that term was introduced by the structuralist Bernard Bloch, who ignored etymology entirely in his analysis. So if you're analyzing Japanese the way Bloch and Jorden do, copula included, I'm not sure if etymology is even a factor.
But hopefully by this point you can see how the words fit together historically, and you can decide for yourself how you want to think about it :-)