Most likely you would have a "Metis" population form and ascend to the ruling class, given that powerful natives in the area often married their daughters and sisters to fur traders for the alliances it offered. They would likely start forming a more centralised, Sinicised ruling class over the course of maybe 250-300 years and would be aided by whatever Han merchants cross the sea to these new chiefdoms. This would be when epidemics start breaking out, and the resulting demographic collapse naturally favours the mixed-race Chinese. Incidentally, inoculation was a known practice in China, so this could make it's way into the region and local medicine. It could be that by the 18th century you have plenty of local states led by mixed-race Chinese, supplemented by emigrants from China and local populations, probably speaking an ATL Chinook Jargon which instead of the heavy French and English influence has a heavy Chinese influence. Although I say "Chinook Jargon" in the sense of its influence in the region, more than likely it would be based on a different language than Chinookan, so perhaps something like a Sinic form of OTL's Haida Jargon or Nootka Jargon.
On the native side of things, the introduction of malaria to the PNW which decimated the natives of the Lower Columbia and Willamette Valley in the 1830s will go differently, since it will be a different strain of malaria and depending on which region of China is trading with the PNW it could take a lot longer for malaria to make the jump. More time for natives to adapt to the epidemic situation and receive Sinicising influence will alter the response to the arrival malaria and save thousands of lives (although many will likely die of other disease).
The wealth brought from Chinese trade would likely start the events of OTL in the 18th-19th centuries in the PNW centuries early, marked by widespread displays of wealth (increasingly grand potlatch ceremonies, spread of totem poles nearly everywhere) and warfare where the northern peoples like the Tlingit and especially Haida and Kwakiutl/related peoples will dominate since they will have more access to Chinese goods and weapons (although firearms won't really be a thing in trade for a few centuries, so this might not be as big as OTL). There will be a lot more forts erected across the region as well. My guess is this warfare leads to the "Metis" gaining influence and eventually forming the basis of states by the late 18th century or so. Probably by this point you'd have real Chinese influence and colonialism in the best parts (southern BC/Washington/Oregon) which would outcompete the lesser, more chiefdom sort of states in northern BC and Alaska.
As for the introduction of horses, this is tricky since Alaska and northern BC/Yukon isn't the best horse country. But eventually horses could make their way south through Chinese and native trade, where they'll inevitably find their way to the Columbia Plateau which is excellent horse country. My guess is that will occur by the early 17th century, so earlier than the Spanish-derived horses. You wouldn't have the "Plains influence" on Plateau Indian culture (for instance, the organisation of warriors, the similar feather headdress and robes found among the Sahaptins, Nez Perce, etc. derive from extensive contacts with Plains Indians). You'd likely see the Kutenai and the eastern Interior Salish people (i.e. "Flatheads" and other speakers of Montana Salish languages) push onto the Plains out of competition with other peoples and opportunities to hunt bison (OTL they were pushed west by the Shoshone and Blackfoot thanks to Spanish horses and guns). It's possible the Nez Perce would push south and take the Snake River Plain from the Numic peoples. I think instead of the Blackfoot Confederacy as a dominant power in Montana and Alberta we'd have a Salish-Kutenai confederation. Speaking of the Plateau, it probably will be dominated by horse tribes and Chinese influence will mostly cease around Celilo Falls/The Dalles. There will obviously be traders/"Metis" people there but no real state formation.
The vast amount of gold and silver will become apparent at some point, but will likely take a few centuries. The discovery of jade will come far sooner, since the Athabaskans exported jade along the grease trails to the coast in exchange for fish oil and other coastal goods. IOTL, Chinese immigrant miners in BC discovered jade and used this otherwise-ignored resource to great effect. The Tsimshian and other peoples of the Central Coast of BC were the main recipients of this trade, so a Chinese expedition this far south will discover this resource. So jade would be the main reason the Chinese would push further south. Alongside jade they would of course trade for furs and antler velvet of elk, deer, and caribou. Should the Chinese go to the source of the jade, they would find the same gold that spurred the gold rushes in interior British Columbia.
I think the Chinese presence on the West Coast would run into the growing European presence starting with the Spanish but subsequently the British and you'd see a lot more European interest there by the 18th century, since this offers the opportunity to trade with China. For the "Metis" chiefdoms and proto-states, this opportunity is huge and probably would be what consolidates them and starts the process of nation building. They possibly might be colonised, assuming China is weak, but might use the competing Chinese, English, and Spanish presence to carve out their own niche. All would be fairly poor and backwards, but with a decent-sized population that they'd never be swamped by European colonists.
Things could go a few different ways from there. Adopting rice farming is by no means inevitable and possibly not even likely, but if it happened it would basically have to be revolutionary in terms of the populations the region could sustain.
Chinese writing does not strike me as a package that would transmit easily through light contact.
Writing is far easier to transfer with light contact, see the spread of aboriginal syllabics in Canada for an example. Although they were invented by a white missionary, the natives rapidly spread them among themselves to the point where they developed their own myth for the origin of syllabics.
Agriculture would be a mixed bag. Many PNW native peoples from the Tlingit in the north to the Salish in the south farmed potatoes, but didn't farm them well since they never used much fertiliser or cleared much land to expand their farms. However, a few species of Sagittaria are farmed in China and apparently, demand can be high enough that West Coast species like
Sagittaria latifolia is harvested and exported to China.
S. latifolia, or wapato, was a major food source for the people of Vancouver Island and especially the Salish, Chinookans, and Kalapuyans who exported it elsewhere. So the Chinese may stimulate more intensive planting and harvesting as well as spreading the plant to places it might grow like coastal northern BC and southeastern Alaska. I'm not sure if there's any analogue to potatoes in 15th/16th century China, since root crops will be easier to introduce to the natives since they already harvested wild roots and semi-domesticated roots like camas. Something like buckwheat, millet, or especially rice would be far more difficult to introduce successfully.
Culture's trickier to talk about: the natives might be interested in things like Chinese religions and philosophies, but they're more likely to adopt the names of Chinese gods into their own frameworks than they are to convert. Certainly stable societies with stable social organizations are unlikely to throw them over wholesale because a few traders dropped by with some shiny beads, but contact with a much wider world than they ever knew existed, and the introduction of new technologies and ideas, would presumably have some effects.
I think it would be very easy for them to equate Chinese figures like the Yellow Emperor or Shennong with Coyote, Raven, or other Transformer figures who created the world as it exists and taught people how to properly hunt and fish. They could also be revered as buddhas. This would be equivalent to how there are Christianised versions of these stories where Jesus (or another Biblical figure) is attributed many of these deeds.
*Worth it* is the question. And I'm not sure that hunting animals for fur--which is about the most primitive economic activity that humans can engage in--is worth regularly crossing the Pacific Ocean for.
The distance between Shanghai and Juneau, for example, is about the same distance as it is from Shanghai to Istanbul. Shanghai to San Francisco? About the same distance as it is from Shanghai to London.
Ships can sail faster than caravans can move, and your only problem is pirates off the coast of Japan. Russia had all of Siberia with its countless furs yet still went to Alaska. I think China would find something similar, especially since the sea otter (whose pelt was long exceptionally valuable in East Asia and attributed magical properties) is only found in Northeast Asia and the West Coast of North America.
there is no need to go all the way to juneau on a permanent basis i agree. However the Chinese did regularly trade all the way up to the Kurils to at times, Kamachatka, so the real question would really only be the distance from the Kurils to Aleutian islands. The Andrean+Aleutian Islands would have more than enough fur for the Chinese to come looking. At that point, the Chinese would be aware of mainland Alaska and BC, but they probably would stick to the Andrean and Aleutian Islands.
Overhunting is a real problem, and the Russians kept pushing east and eventually established bases on Kodiak and subsequently penetrated the southeast of Alaska. I don't see why that wouldn't eventually happen here.
