1. Do we really need an explorer/expedition type vessel for a doomsday scenario? "Explorer" type is just marketing speak for having some sort of ice rating, meaning the vessel can handle herself top of 66 degrees north and bottom of 40 degrees south. Since these are some of the most treacherous waters on the planet, won't attempting to navigate them in times of disaster essentially be out of the frying pan and into the fire? I really can't imagine any doomsday scenario where we might want to go there especially since getting a vessel an ice rating is a significant expense.
2. I am still not sure why relying on an regular crew is a good idea here. Doomsday/SHTF implies resources you're used to are no longer available. So, what if the engineer shows up for work, but the skipper doesn't? What if the crew decides the owners are expendable and take the boat for themselves? Also, this ties into the expedition nature as well since captains/engineers who have ice experience are pretty rare. I believe the first criteria for a doomsday boat should be that only the owner can access the vessel and needs no one else to operate it.
3. Shouldn't we be trying to downsize the vessel to ensure that it remains manageable? Larger vessels require more upkeep than smaller ones, and have more things that can go wrong, after all.

You appeared to be confused about volcanoes, wondering why they would all go off at once. That is NOT the concern. Preppers are concerned with SUPERvolcanoes. Supervolcanoes can expose hundreds of square miles to open volcanic activity, burying a continent in thick layers of ash and creating winters that go on for years. If the Yellowstone supervolcano went off, it would, for all intents, destroy the US and Canada's ability to produce food for many years, in addition to the more immediate effects. (We are technically "overdue" for a Yellowstone eruption, by the way.)