How many ethnic Russians, or inhabitants with recent roots in Russia, were there in Alaska when US bought it? Were they hunters, farmers, traders? How many stayed and became Americans?
The Wikipedia article on the Alaska Purchase discusses the population of Alaska before the purchase:
Seward told the nation that the Russians estimated that Alaska contained about 2,500 Russians and those of mixed race (that is, a Russian father and native mother), and 8,000 indigenous people, in all about 10,000 people under the direct government of the Russian fur company, and possibly 50,000 Inuit and Alaska Natives living outside its jurisdiction
This population of Russians were part of the Russian American Company, which was mainly invested in gathering furs. The treaty of the transfer allowed for any Russian who wished to remain to to so.
From History of Alaska by Bancroft:
...treaty provides that with the exception of the uncivilized native tribes the inhabitants of the ceded territory shall be admitted to the enjoyment of all the rights advantages and immunities of citizens of the United States and shall be maintained and protected in the free enjoyment of their liberty property and religion
These individuals, however, were essentially homeless and jobless in what rapidly became a classic 'frontier' region, and with many disadvantages.
The natives were not slow to take advantage of their opportunity and refused to sell the Russians game or fish at former rates while the Americans refused to accept the parchment money which formed their circulating medium in payment for goods except at a heavy discount
So most were relocated by the Russian American Company. Again according to Bancroft:
Within a few weeks or perhaps months after the transfer there were not more than a dozen Russians left at Sitka the remainder having been sent home by way of California or round the Horn
Apparently, the vast majority of Russian settlers went home. However, some remained and preserved their culture. Several years ago, I've read an article about a peculiar dialect of the Russian language which managed to survive in a remote Alaskan area, a village named Ninilchik.
Russian sources (https://www.gazeta.ru/science/2013/07/09_a_5417937.shtml) indicate that the village population never exceeded 200 - 300 persons.
Here's a couple of English versions of the same story:
The settlement was founded in 1847. Ethnic Russians who had settled in Alaska and intermingled with the locals were much of its population. When the Russian Empire sold Alaska to the United States in 1867, links between Ninilchik and Russia broke off. Up until the 1960s, the locals had next to no contact with other Russian speakers.