Sahaidak, being (my assumption based on his nik) from x-USSR and familiar with Russian version of history, got it right. Furs were THE reason for Russian colonization and one needs to take fur trade out to made Siberia undesirable for Russia initially. How to do it, I have no idea. Rabidly anti-Russian Poland, may be, permanently at war and blocking trade routes with Europe. Plus Ottoman Empire shifting a lot of their attention to Russia from Balkans.
Keep them out of Kazan. They started expanding east once the khanate became a Russian subject under Ivan the Terrible. More to the point, the expansion was directed largely from the upper Volga. If the Russians aren't on the Volga, they'll probably do little more than Novgorod did in OTL - loose sovereignty over parts of western coastal Siberia.
Upper Volga was under Russian control before Kazan fell, so this route to Urals remains open even if Kazan remains independent but does not expand significantly.
Someone here once wrote that Sibirian was colonised because Russia tried to control the Cossack, and they chose to flee into the Sibirian plains but the Russian state followed them the entire way to the Pacific ocean.
Relationships between Cossacks and Russian govt were complicated and could not be described in "war" or "serfdom" templates. Closest analogy would be relationships between successful Carribean pirates of early 17th century and English Crown. Gangs of outlaws, strongly opposed to govt control but often ready to share their spoils with govt in exchange for forgiveness or protection. Cossacks "bowed to Czar with newly aquired lands and it's inhabitants and asked for forgiveness". Yermak himself had been an outlaw for a long time. And "Cossack States" in Siberia is ASB. Cossack units rarely exceeded several hunderds men (pretty often being several dozen strong). There're no chances to create "STATE" with population this numerous.
There were a lot of people that came to Siberia to avoid control but those Cossacks that expanded Russian influence were in service to Russian state (the other arrived later).
And even outlaws were always willing to make a deal.
Although Russia had pushed beyond the Urals during Ivan the Terrible's time, the real push toward the Pacific did not happen until the mid-17th century during the reign of Alexis I.
For the bulk of Siberia to not come under Russian rule, the "Time of Troubles" (1598-1613) between the fall of the Rurik dynasty and the rise of the Romanov dynasty would have to be a lot more troublesome and long lasting. Either that or Russia comes under the rule of a foreign king, i.e the king of Poland.
AFAIK Russians reached the Pacific in 1640's, first fights between Cossack and Chinese (or Chinese clients) are mid-1650s. And Russia under Polish king could be even more expansionist (IOTL decade or two had been spent to re-establish national administration, in Polish Russia it would likely be Polish administrative model quickly expanded in Russia).
Because they showed
so much interest in the time before the Russians came...
Settlement activity in Siberia is ASB before mid-19th century, as there's no type of agriculture viable in this climate. Even today agriculture remains confined to southern Siberian borderlands. Chinese do not have type of agriculture suitable for conditions (unlike Russians, who could and did transplant their ancient "rye and cattle" model in Siberia).
Edit: Answering OP question, I see some form of "Scramble for Siberia" in mid 19th century, similar to "Scramble for Africa" IOTL. With same participants, plus Japan (likely) and China (reluctantly, but they will be drawn in the game too).