Reply of the Zaporozhian Cossacks

Turkic origin is not the only possible root - there is also a Scandinavian theory that ties hetman and otaman together (their meaning is similar and at some point was used interchangeably) and also a Germanic root.

Source is in Russian (couldn’t find an English one): https://ru.m.wikipedia.org/wiki/Атаман

The 'Cossack' Wikipedia page explicitly says that 'Kazakh' isn't to be confused with 'Cossack.' Are they in fact related etymologically?

One can say he did not, and they did. The most likely context of this is one of the countless wars between Ottomans and Russia for the control over Ukraine, south of Russia and Crimea. Most likely this one: https://en.wikipedia.org/wiki/Russo-Turkish_War_(1676%E2%80%... Note that there were different Cossack communities there, and some were recognizing Ottomans as rulers, and some did not. The initial letter was probably written to the latter, and is by the standards of that time pretty mild - basically just asking the Cossacks to bend the knee, like their fellow people on the other end of the river did, and stop harassing the Ottoman territories. But these Cossacks were having none of it, and by the end of the war Russia ended up owning their side of Ukraine (separated by the Dnepr river) while Ottomans ended up owning the other part. Of course, it soon led to the next war, of which there was about a dozen overall. Cossacks as a community (or rather very diverse set of communities) survived most of them, until the Communist revolution, which largely destroyed their way of life. Though it is said to be reviving now, not sure whether it is authentic anymore.

One interesting detail is the rolled banners. Some of them are yellow and blue - same flags that modern Ukraine uses. Others are black and red - the colors used by Ukrainian rebel armies against Polish Lithuanian Commonwealth, Ottoman Empire, and Russia. Centuries later black and red banners were used by Euromaidan protesters.

I've always interpreted it as capturing the moment the 'kill a hedgehog with your bare ass' line is being composed.