Having gone to school in the Pacific Northwest, with a teacher who very much loved rocks, this was actually something we covered in class with documentaries, etc.
What are the largest geographical implications?
Yellowstone would flood a very large area in lava rather quickly creating some lovely plains. The earthquake would probably trigger our San Adreas fault which would set off pretty much every fault in California as well as causing the Juan de Fuca Plate to subduct. The resulting quake would have three major effects and possibly a fourth. The first would be that it would probably set off volcanoes in the Ring of Fire (not immediately but as seen on a time lapse of recorded volcanic eruptions would look pretty instantaneous). The second would be the strength of the earthquake makes it behave completely differently than normal and "earthquake-proof" structures would receive much more extensive damage than regular wooden houses, large towers would be gone. The strong quake would induce liquefaction in extensive areas west of the Rocky Mountains so places like Puyallup would disappear underground and events like the Oso mudslide would happen in many areas. Third, the tsunami generated would be massive and at least 80 ft high, probably much higher. The tsunami would come after the earthquake so the structural damage would probably allow the tsunami to collapse any remaining structures. You have typical flooding problems and slightly more drastic tsunami ones around the entire Pacific.
The fourth is it could trigger Rainer which would be much worse. Also Mt. St. Helens if its re-pressurized enough and the eruption of Rainier doesn't effect it. If it did trigger it would be quicker than the Ring of Fire. Any hope of anyone surviving west of the Rocky Mountains probably just completely disappeared. Ash levels would significantly increase especially in our bread-basket states.
Would humanity would survive such an event?
At least some of humanity would survive pretty much anything that could be dished out from such an event, especially around the Atlantic. The different regions would be affected separately and at different times and we know enough to be able to have at least a small group dodge most of it.
Extent of damage to nature both in the ash-covered areas and the surrounding landscape?
How long would/could it take for the winter to lapse?
I don't remember enough about the expected ash levels to say. It definitely would release a decent amount of particulates into the air, but going off any other high-ash events I could think of we'd be fine (the world won't turn into Pompeii and it would certainly effect temperature but I don't think we'd get too cold).
Does it get worse?
You didn't ask about non-geographical but its probably worth mentioning that a large part of the global economy is centered around the Pacific, that the US provides a lot of the foreign aid in terms of disaster relief, and that we would have a large drop in US food production. You would probably see the worst global depression unless the death-toll offset the gross losses in production, neither being a great scenario. Also you should probably count the number of nuclear reactors and radioactive dump sites around the Pacific Rim and add them all to the "apocalypse-level" around the Pacific.
Note: The chain reaction is an extremely likely scenario if Rainier were to go off first. Please note that the chances are lower for the chain if other things go off first. The lowest chances probably occur with Yellowstone erupting first, but its still a possible scenario.