score:3
I went to Moscow last summer for 10 days. I'm a student so I had to make sure it was the cheapest possible. I'm also from France so it wasn't entirely difficult to go (may be different for your situation).
You want to stay for an entire month is kind of a stretch, seeing that you are not "rich
" (very vague, what's your budget btw?).
Getting an invitation to Russia requires having a hotel specified, but once you're in Russia you can choose anything you like, as long as the Russian governement knows where you are (if 7+ business days)
I must get an invitation letter and voucher in advance to apply for a visa at the local consulate. However, while the hotel issues them, this may not be true in the case of AirBnB.
This is entirely dependent on the place you wish to stay, you need to contact them and see if they offer invitations, which is highly unlikely. Invitations are usually done by Russian touristic companies. There are different kinds. I went through VisaToRussia.
The only downside is that they make you pay 40$ for the invitation, but at least you won't have to break your head open trying to find a fake hotel and then switching once you're in Moscow.
So to travel for Russia, is it possible to use AirBnB (possibly all rooms, not just those rooms offered by business organizations, which is true these days around the world)?
Yes.
is it easy enough to register my visa by myself?
I don't think this is possible.
register my visa for immigration
Most services like VisaToRussia do offer the possibility to register your visa but for additional fees.
If that is not an option for you, you need to contact the place you wish to stay at to see if they do know how to register your visa.
Or may I apply for another type of visa to stay in AirBnB if at all possible?
You could come to Russia via another type of Visa, do you have a contact in Russia that could invite you to the territory? (However, they would be entirely responsible for you.)
While this may not completely answer your question, I felt the need to point out certain information. Cheers.
Redacted invitation and voucher from VisaToRussia
Redacted invitation and voucher from VisaToRussia