Is it true that Americans think about Russians, that they walk bears instead of dogs?

It is not true. Actually ask an American and you will find this out. 

No.  Americans very obviously know a lot more about "Russians" than you know (and keep proving!) about the USA.

When I think of Russia I think of extreme poverty. I think the people that live there must be extremely dumb to allow themselves to be treated that way.

Actually I think that bear thing about Canadians. Russians probably just take turns walking the town cow, so it doesn’t freeze to death in the summer. 

I didn't think about Russians until Trump used them to steal an election.

They indeed do, the little ones. 

I have a different stereotype image that pops into mind when I think "Russians". I see  an intellectual philosopher reading Dostoyevsky by a fire with snow outside the window and sipping on vodka.

Yess, ....... FACT , ....