Did the Red Army use tactics from Finland against Germany ? I searched for a while and I have never found any trace of it ...
Yes. It gave them just enough time to prepare to stave off an even larger disaster than what happened in Summer 1941 when the Germans invaded. It's hard to imagine a modern army more inept than the Soviets in Summer 1941, but it was the Soviets in Winter 1939.
The Winter War humiliated the Soviet Union and its military. It showed that their military leadership was inept, their strategies were crude and ineffective, and their intelligence organization was full of yes-men. As a result, the Soviets began to undo some of the damage of the purges. The changes were fundamental, applicable to attack or defense. The Soviet army which cracked the Mannerheim Line in early 1940 was already a more professional army than the one which bumbled into Finland in late 1939 with self-delusions of being greeted as liberators.
William R. Trotter writes in A Frozen Hell p 264...
...the Supreme Military Soviet had met in urgent conclave in April 1940 to sift through the lessons of the Finnish campaign and recommend reforms. The role of frontline political commissars was considerably reduced, and old-fashioned ranks and forms of discipline were reintroduced. Clothing, equipment, and tactics for winter operations were thoroughly revamped; so were tank and aerial tactics. Professionalism, in short, was put back in its proper place. Not all of the reforms had been completed by the time the Germans struck, but enough changes had been wrought so that the Red Army was a far tougher and better equipped opponent than it would have been had the Winter War never happened; there was at least a thin margin of improvement that enabled Russia to survive, just barely, the stupendous onslaught of the world's finest professional army.
The Winter War provided a clear and relatively cheap lesson for the Soviets: they had to change and change fast. Had they not begun those changes in 1940, better training, professional military leaders, better winter equipment and training, the Germans may well have entered Moscow and Leningrad in winter 1941. As poorly as the Soviets did, they were able to stall the Germans enough to throw off their very optimistic timetables.
Still, this didn't stop yet another purge of the Soviet military leadership before and during the German invasion.
One of the clearest outcomes of the Winter War was to increase production of the then brand new T-34 tank at the expense of light tank production. This allowed the Soviets to rapidly replace their early losses of obsolete light tanks in 1941 with the far superior T-34. Without the Winter War, they would have been replacing their light tank losses with more light tanks.
Prior to the Winter War the Soviets put their faith in light tanks, such as the BT and T-26, or overly complicated tanks such as the T-28. Their light armor made them very vulnerable to even the small number of relatively light Finnish anti-tank weapons.
Soviet tank crews were poorly trained and poorly organized. They had a tendency to charge ahead of their supporting infantry, break through the lines, and then sort of mill around waiting for orders which weren't to come because they often lacked radios. Without supporting infantry (often busy being mowed down by Finnish small arms fire) they were easy pickings for Finnish anti-tank teams armed with explosives, Molotov cocktails, and anti-tank rifles.
The Soviets learned from this. They needed tanks with heavier armor. The tanks needed to coordinate with the infantry. They needed radios. And they needed more tactical flexibility. All the basic reforms the Germans had been training and practicing for years. These reforms were in progress when the Germans invaded. They gave the Soviets a year head start on the learning game.
The T-34, without which, it can be argued, the Soviets could not have won WWII, saw its combat debut in the Winter War. The then brand new, relatively expensive, and controversial tank showed its stuff. With the poor performance of light Soviet armor this ended the debate over whether the Soviets should use many cheap, fast, light tanks, or fewer more expensive medium tanks. Production of the T-34 would be given priority in September 1940.
No it didn't. If you take a look at initial Soviet losses against Germany, you will see that basically all (approx. 100%) of the front-line forces were lost before October 1941, mainly in encirclements. Potentially, you could try to find one or two veteran divisions that were held in reserve and deployed for the winter, but from my memory I think there were no such cases (there could be division re-numbering, but not the actual people moved).
There is no evidence that any part of tactics which Soviets used in 1943-1945 was developed specifically in the Winter War 1939-1940.