From the main window "My Library", Better BibTeX adds the citekey at the top of the metadata, in the Info tab of the right-hand side Item Pane, as Citation Key: ---.

When you open a PDF file in a Zotero PDF Viewer tab, you also get the metadata in the Info tab of the right-hand side Item Pane. But that Info tab does not seem to inherit the same settings as the Info tab from the "My Library". For example, the font size in there is different:
https://forums.zotero.org/discussion/98932/shortcuts-for-changing-the-font-size

So the citekey display disappears from the Info pane when you are in a PDF Viewer tab.
I think he is asking to keep the citekey display feature in the Info pane visible on the side of a PDF file viewer tab. I also think it would be useful to keep that Info tab consistent between My Library and the PDF Viewer.

Yes.

Please open an issue for this on github, that's how I track requests. I had totally forgotten about this.

Yep forgot about that one too -- but joining that issue or even opening a new one is preferable to "+1"ing things here.

I'll get to it eventually, but I need to sort out some prep for Z7 first.