Today I got so annyed by firefox's default behaviour of downloading each and every PDF file to my disk that I went searching for a solution until I had the problem fixed. And it seems like I have finally found it. I have linked the solution but here is the fix in short:
go to about:config
change browser.download.open_pdf_attachments_inline to true
it isn't in right in firefox's settings - general - under 'files and applications' ??
'open in firefox' (what you want), vs 'always ask', 'always save' (download) or 'open in default application' (such as adobe reader). actions are configurable for many file types.
I believe even if you choose “Open in Firefox”, it will still download the PDF to the default download directory before opening it inside a Firefox tab. The behavior that OP describes above seems to prevent that downloading (and having the file around in your default download directory).
I don't know whether they fixed it since, but last I checked that option was broken.
No matter what you select, half the time it still downloads the PDF to your drive.
It even says in the description that these settings determine how firefox will handle them after downloading them.
i have 'open in firefox' enabled on this pc for pdf. i never get a dl prompt. goes right to the internal pdf viewer every time.
The problem is not that it gives a download prompt, but rather that it tosses the PDF file into your downloads folder unrequested.
It opens the PDF in the internal PDF viewer as well, but that is not the thing people are having issues with.
NONE of the files i tested it on triggered the download prompt, populated the download status of the browser, or appeared in my default downloads location during their 'viewing' or after.
Great that it seems to work for you, but I've been experiencing this bug for months now if not longer.
I made the changes OP suggested, and now it seems to work correctly.