Here's my take :
On Gmail you can check the source of a mail. Thats gives you the source HTML. With Google-sheet or python or whatever you want, just extract the links and you're done.
Concerning this issue : Is it not possible to print a pdf from a hyperlink?
Here are relevant answers :
Many of today's printers support direct PDF printing. Lexmark, HP, Xerox to name a few all have this on most of the 'business' printers. On these devices simply sending the PDF file directly to the device over LPR, port 9100, or some other mechanism will result in a printed document. Some devices even support URLs. I do know that Lexmark had some devices that a URL could be sent to the printer as as long as it had access to the URL it would pull the document and print. In this case it supported basic HTML, JPEG, TIF, and PDF.
A PDF must be rendered as an image before it can be printed. Usually when you're printing a PDF file on your desktop you could simply right-click on the file and select Print and if you have Adobe Reader or an alternative application set as your default PDF viewer, then the PDF that you have selected will be opened automatically -- at this stage the PDF is rendered as an image -- and then the printing process will begin.
But if there is no access to a PDF viewer that can render the PDF and then print it, then you won't be able to print the PDF. Usually if you have Adobe Reader, Foxit Reader, etc, installed then when you click on a URL to a PDF then the PDF will open within the PDF viewer within the browser and you will be able to print it.
Alternatively, you could find a PDF SDK that silently renders a PDF as an image and then sends that to the printer, without the need to have a PDF viewer installed on your machine.
Credit to the orginal authors.