Napoleon and Hitler most likely had better organised armies, and though the Mongols were more northern in nature (so more accustomed to such weather) how did they manage to survive when Napoleon and Hitler couldn't?
NEITHER, repeat, NEITHER Napoleon nor Hitler invaded the Russian Empire or Soviet Union respectively, in the winter. Napoleon invaded June 24th, 1812, Hitler invaded June 22nd, 1941. The Mongols were faced with a less numerous, much less organized foe than either Napoleon's or Hitler's armies were faced with.
You are placing a lot of emphasis on the fact that the Mongols were successful because they could survive the Russian winter. That assumption is not entirely correct. The Mongols were successful because they used superior methods of warfare and because they were good at subjugating defeated combatants. Let me preface this by saying that I am not a Russian historian, but I have tertiary knowledge of the subject.
Structure of the Rus' at the time of the Mongol Invasion
In the context of the Mongol invasion of Russia, one must see that Russia was not very large. Russia was actually made up of the Kievan Rus'. The Kievan Rus' was government made up of several principalities. The basic structure was the Grand Prince of Kiev controlled all of the lands around the city and his relatives controlled the surrounding cities and subsequent lands. These relatives paid tribute to the Grand Prince of Kiev and we, in effect, his subjects. The Kievan Rus' were politically connected to Western and Eastern Europe. Prince Yaroslav married his sister and three daughters to the Kings of Poland, France, Hungary, and Norway in the 1030's. Prior to the Mongol invasion, the princes of the Kievan Rus' fought among themselves and were already a weakened state.
The Republic of Novgorod was part of the Kievan Rus', but it was governed by a local assembly and all government decisions were made by that assembly. A prince was elected as the military commander of the city. The regions of the Kievan Rus' were: The Republic of Novgorod, Vladimir-Suzdal, Smolensk, Polotsk, Murom-Ryazan, Chernigov, Turov, Novgorod-Seversk, Volhynia, Kiev, Pereyaslavl, and Galicia.^1
The politics of the Kievan Rus' are obviously more complicated than this, but again, I am not a Russian Historian. I'm sure that someone else could give you a rundown of the political landscape before the Mongol Invasion.
A map of the Kievan Rus' principalities from the 12th century might help. It was taken from the lecture notes for Eurasian Studies 201 at Western Washington University.
The Initial Invasion
In 1236, an army of 150,000 Mongol warriors invaded the territory of the Kipchak Turks. You can find the Kipchak region on the map referenced above. Sübedei, arguably one of Genghis Khan's greatest commanders, and Batu, son of Jochi and grandson of Genghis Khan, led this campaign. The Kipchaks fled to Hungary or were absorbed into the Mongol army. In 1238, the army advanced on the Kievan Rus'. In his book The Mongol Art of War, Timothy May notes that it was the winter of 1238 and the Mongols used the frozen rivers as ice roads to get to the Rus'. He only mentions this fact and does not elaborate that it was a key factor in the defeat of the the Rus' principalities. The Mongols defeated city after city until finally they took down the city of Kiev. The invasion and conquest took only 2 years to complete. This was a relatively short amount of time for a conquest.^2
Tactics, Warfare, and Organization
May mentions that the disunity of the Kievan Princes contributed to their defeat. The organization o the Rus' army was quite simple. Each principality has its own army for defense, which was led by the respective prince. Each army had three levels of warrior. The druzhina were the elite fighting force. They served as heavy cavalry and possessed armor. The smerdy were the peasants who formed most of the infantry. The infantry was supplemented with mercenaries. The third level of warrior were the nomads. These nomads were mostly Kipchak Turks who had fled from the Mongols in the years previous. They ranked close to the druzhina in fighting ability and usually fought alongside this division.
In a previous question, How did nomadic societies like the Mongols and Turks reach a technological level sufficient to conquer agricultural societies?, I talked about Mongol military tactics which were employed at the time of the invasion of the Kievan Rus'. I won't copy and paste from that answer, but please feel free to read it for more information regarding Mongol military tactics.
In addition, the Rus' were not properly equipped to fight the Mongols. Only the druzhina had proper armor. Most of the infantry were susceptible to Mongol arrows. The army was also lacking in discipline and lost their nerve when the Mongols led a charge. May notes that the Mongols would ride in complete silence while charging and break out into a blood-curdling scream just before making contact with the opposing army. This would break the will of anyone not adequately trained and lacking in discipline.
Outcomes - The takeover of the Golden Horde
After the invasion of the Kievan Rus', Sübedei and Batu continued their campaign and invaded Hungary. Most historians argue that they would have continued through Europe, but the death of Ögedei Khan in 1240/1241. The army had to turn around and make their way back to Mongolia to choose a successor to Ögedei. After the successor was chosen, Batu took his portion of the army and essentially created the Golden Horde.^3 Honestly, it is, again, more complicated than that but I don't think you were really asking about the Golden Horde. You were actually asking about the Mongol conquest of Russia and why they were successful but others were not.
^1 Russia: A Country Study, Published by the Library of Congress Research Division.
^2 The Mongol Art of War, by Timothy May, 2007.
^3 Genghis Khan's Greatest General: Subotai the Valiant, by Richard A. Gabriel, 2006.
Additional other bits came from:
Luc Kwanten's Imperial Nomads, 1979.
Michael Prawden's The Mongol Empire: It's Rise and Legacy, 1961.
hi! hopefully one or more of the Mongol or military historians drops by to help you out, but I've found a few similar previous posts that might get you started. The answers aren't necessarily in-depth or well-cited so take these as possible leads that need follow-up
Meanwhile, if you want to dig more deeply into the Napoleonic & WWII invasions of Russia with regard to winter, check out these posts (a few contain links to several more):
Did Napoleon and the Nazis face winters of historic harshness in 1812-13 and 1941-42?
Did the winter/conditions really make the Germans lose at Stalingrad
How important was the cold to the defeat of the Nazi war machine in the Soviet Union?
What are some main reasons that Operation Barbarossa failed other than the Russian winter?
Why did the Germans invade the Soviet Union without adequate preparation?