After the 'land bridge' across the Bering strait was reclaimed by the sea c. 8,000 years ago, is there any evidence that Native American 'Eskimo' and Native Siberian Chutchka, Evenk or Yakut (or other) peoples had any contact (genetically, culturally, or via trade)?
I'm not in arm's reach of credible printed sources at the moment, but if the moderators will be generous to me I'll use reputable online sources to make one or two 'common knowledge' observations.
I have to point out that the "Eskimo" groups you're thinking of (called "Inuit" in Canada and elsewhere) are actually the descendants of a colonizing culture, the Thule, that arrived in North America in the 9^th Century across the Bering Strait. They spread across the Arctic and completely displaced the previous dominant culture, the Dorset, by the 15^th Century.
So, yes, there is indeed evidence that aboriginal peoples had contact across both sides of the strait after 6000 BC - but not in the way that you envisioned, since the Inuit are not themselves aboriginals from North America. As late as the 15^th Century, "colonizing" Inuit peoples were still displacing "aboriginal" Dorset peoples in remote regions of what is now Nunavut, Labrador, and northern Quebec.
Here is a map of Arctic cultures from AD 900 - AD 1500.
