Note that the following assumes that all players are playing a perfect game with no mistakes. In real games every player will make occasional mistakes so take what I'm about to say with a grain of salt. For the details on optimal play check out http://donsessays.freeservers.com/
The Russia first turn restriction does nothing to change the balance of classic Axis & Allies. This is because Russia should never attack on the first turn. A&A classic offers very little margin for error in its economy. If Russia produces 9 infantry a round it will survive, if it produces 7 a round it will die. In order to keep up production Russia must conquer Norway during the second round. No other ally may take Norway; it must be Russia. It can't do this if Karelia falls. The only way to prevent Germany from taking Karelia on the very first turn is to place literally all of your defenses in Karelia except 1-3 infantry in the Caucasus (to prevent a blitz through to Russia) and 7 infantry + 1 tank in Yakutia (you can't defend the Russian Far East at first so don't try.) Attacking Manchuria on the first turn will cost you the game. You must slowly build up a large stack of infantry in the Yakut SSR to prevent Japan from breaking through.
On the other hand extending the restriction to turn 2 would mean certain doom for the Allies. Yakutia must pose a second turn attack threat to Manchuria so Japan reinforces it instead of throwing everything at India. Russia must also be able to take Norway and Ukraine on turn 2 or else it won't have the money to crank out 9 infantry on turn 3. Eventually the US will take pressure off and you'll have more of a margin for error, but the first five or six turns are critical.
This house rule might work in some other edition of Axis and Allies, but in Classic it would break the game. The 1-turn rule does nothing but prevent Russia from killing itself prematurely, but more than 1 turn would change the game fundamentally.
A two-turn restriction would not make sense from a historical standpoint because the beginning of the game represents a moment when Russia and Germany were already at war. You'll notice that the Ukraine Soviet Socialist Republic is already occupied by German forces. In real life the USSR did not go to war with Japan until three weeks before Japan's surrender. Of course in real life Japan never posed a real threat to the USSR, and they certainly weren't capable of marching across Siberia and attacking Moscow from behind.