there are hostels in Moscow and St.Petersburg. the price is about 700 rub. Общага - is different, it's not a hostel (it can be even 300-400 rub, but because of this cheapness it can often be illegal). Hostel are the same as hotels, but a bit cheaper. I don't know much about the registration rules, if you travel as a tourist you will logically have no need to register. I know that when my friend invited foreign friends as guests for 1 or 2 weeks in her home she had to register them (and it was a stupid bureaucratic procedure). But if you don't have a host that invites you, why register? just live in hostels or hotels, be prepared that people don't speak English in the street, and it will be difficult to travel from place to place without good knowledge of Russian. Why ask about bribes if you are not going to do anything prohibited?