Is it possible to configure the browser so that after downloading the pdf, the document opens in a new tab?
You need to go into Tools -> Settings -> General and set the Action for PDFs to Always Ask. It will then show you a dialog of what you want to do. If you choose Open in Firefox it will open it in a new tab. It will save it in whatever you default download place is. If you set the Action to just open in Firefox, it will open in the same tab but view it online without actually downloading it.
Now, if I do keep the Action to Open in Firefox, I can right-click on the link to the PDF and choose open in a new tab.
Anyway, that's how it behaves for me.