That answers my first question
Well since you are to make money in Moscow and not to spend it(I mean you are not on vacation) you wouldn't be able to live in a hotel for a long time, you earnings would not cover it, I think. So you would need to rent, renting in Russia is somewhat a problem, especially if you need an affordable place to live.
Russia also requires you not only to have a visa but also to have a registration at the place of living (a special paper or mark in a Russian passport) that would tell that you actually live at a local address. Hotels provide such paper but if you are to rent a place from a person that person might be reluctant to provide you such document
Don't get me wrong. Moscow is a beautiful city it is just very expensive and private tutor earnings are not great. Think of it as you would need be able to live in New York city on a salary of a social worker.
PS: I don't know what most are saying but if any of those most actually started a life in another country then listen to them if they don't have such experience then the value of their opinions should not be very high for you.