Hello,
Yes, it's the original document that's required. Currently, allow between 4 and 7 weeks to receive it via Russian post. It doesn't seem like an electronic version exists, and it's certainly not sent to the Russian consulate in France.
You can apply for your visa through the embassy using an electronic form that you'll need to print. Don't forget that you must have valid repatriation insurance for Russia (ask for the word "Russia" to be specified on your insurance, and you need the original—insurance companies usually know how to do this). You'll also need valid photos; the requirements are on the Russian embassy's website.
You can submit your application with a copy of the invitation to the embassy, explaining that you'll receive the original later. It will be required when you go to pick up your visa. (I've done this myself)
With a private visa, you have three working days to register with the UFMS (Foreigners' Service), unlike a tourist visa, which gives you seven days. It's mandatory!
You'll need to go, with the person who issued the document, to the UFMS to register (it's free). Don't do it via post (it's paid), and documents can sometimes get lost.
Best regards,
Patrick
(By the way, Russia is to the east—could you specify a region to estimate the postal delivery time?)
Un concerné n'est pas obligatoirement un imbécile encerclé