So Nikolai Rhyzkov wouldn't stand a chance?
Nope, he was a Commie candidate and a Commie had as much chances to get elected in 1991 Russia as in 1991 France (i.e. not many).
Who would be Sobchak's running mate, Rutskoi, Primakov?
Hard to say. Rutskoi was star veep candidate due to his stellar war record, shielding democrats from accusations of being unpatriotic puppets of foreign powers. He might very well to pop up as Sobchak's running mate. I'm dubious about Primakov. He might become Premier, but not largely ceremonial veep.
So you envision a rump USSR consisting of Russia, the Ukraine, the 'stans and the Caucasus?
I'm not saying it is a given, but Belovezha agreement was at least as much fueled by Yeltsin's desire to pull carpet from under Gorbachov's throne as general dissatisfaction with Communist rule. It is possible that Sobchak would not advocate independent Russia as the only way to get rid of communist tyranny and would work with Gorbachov at reforming USSR into democratic country with mixed market economy instead. Georgia could be a sore spot of this "New Union", so it might be allowed to let go. It is of no interest to Union government holding Baku and Yerevan (with benefit of hindsight I would suggest that it would be used as "failed state" bogeyman by Union authorities; something like "would you continue to play your silly separatist games we would kick you out and you would be as sorry wasteland as Georgia").
According to wiki, Sobchak was the mentor of Medvedev and Putin, would he rule like a Putin?
Sobchak could be called "Medvedev's mentor", but Putin had been grown up man when he had been invited to join Sobchak's team in Leningrad to act as liason between democrats and law enforcement. So, I think that drawing any parallels between Putin's style (be it good or bad) and Sobchak's ideas is pointless. Medvedev's noises about reigning in corruption and creating lawful state in Russia, however, could be at least partially attributed to Sobchak's influence. After all, the guy was a law professor all his life.
Hard to say, but it is doubtful that situation in 1991-1994 could be made worse there, so any changes probably are to the better (and I consider law and order "better"; Chechen independence movement run exactly opposite to "law and order" paradygm). At very least, IOTL run of Soviet Army from Chechnya on Yeltsin's orders might not happen ITTL, leaving separatist gangs (forgive me for using politically charged language, but I do have problem calling groups of ethnic cleansers any politically correct term) there lacking arms needed to resist regular army (analog of 1994-1996 war IOTL).