I ask in regards to military strategy and the like. I read the story today about Norway supposedly upping it's border security against "unpredictable Russia" and it got me thinking about Russian winters vs. Canadian / American winters. Obviously it's further north than most of the United States, but would say a ground invasion of Russia be difficult for America? The only instance I can think of is Germany during WWII and the adage goes 'Don't invade during a Russian Winter.'
Has Russia ever been successfully invaded besides a couple edge territories / border cities / land?
"Russian Winter" is just shorthand for a variety of factors. The climate itself is rough, Russia is of course colder than France, but it's nothing insurmoutable. For example, the average winter temperature in Moscow is about -4C, compared to -1C in Chicago. The emphasis on weather, "General Winter" and all that is just a good mental image, there wouldn't be anybody living in Russia if everybody died in the wintertime.
To that note, Russia has indeed been invaded successfully several times in history. A particularly notable example would be the Mongol invasion of 1223, wherein the Mongols razed most of Russia's largest cities and took over large swathes of territory. The resulting Mongol state known as the Golden Horde existed in Southern Russia for 300 years afterwards.
Besides that, Sweden, Poland-Lithuania, the Ottoman Empire, all fought wars in Russia and seized Russian territory at some point in time. Before Ivan IV created a singular Russian state, the Russian princedoms fought many wars against each other. Russians are not supermen powered by their native soil.
So while the two most famous invasions (1812 and 1941) of Russia did end in many soldiers freezing to death, they weren't so much undone by weather but by compromised supply lines. Napoleon's armies fed themselves basically by stealing supplies from the countryside. The Russians however, refused to engage his army and instead burned as many of their own villages as they could. Napoleon entered Moscow in the middle of winter, without food, which then forced him to try and retreat. That the winter turned out particularly cold was no help, but Napoleon had lost regardless.
Nazi Germany was in a comparable situation in 1941. The Germans reached the outskirts of Moscow (It has to be said, this doesn't mean they were capturing Moscow), but had outrun their own supply lines weeks before. Besides the winter clothing, food, ammunition, fuel, and soldiers themselves were only arriving in a trickle. With no supplies, the Germans had no chance of capturing and holding the city.
Despite all this trouble, and the especially cold winter of 1941, the German survived the winter by retreating from Moscow, and after building up more supplies and reinforcing their armies that launched new offensives in 1942. Again, the weather itself is not the complete story.
As for America, the US army is in no position to invade Russia. On the other hand, Russia is in no position to invade the US. Both countries have similar advantages, a vast expanse of water and/or land separating them from their enemies. Neither country has structured their armed forces around invading the other, so the question is bit like asking how well you can drink soup with a knife.
One thing to keep in mind is that it is not necessarily the cold that is the problem. The bigger issue is mud. Before modern times (and probably still now on the worse roads) the autumn rains and spring thaw would turn all the roads into mud. This means it is incredibly hard to move your supplies, let alone heavy metal cannons.
The cold wouldn't be a big issue if the invaders would have packed the right supplies and clothing, which as far as I can remember didn't happen in either case. The mud slowing down the advance gives ample opportunity for harassment though, which the Cossacks in particular were very good at in Napoleon's case.