Certainly for the mountaineering part of this trip the Arctis Expe ($350) would be a good choice. They聮re a warm, plastic mountaineering boot that聮s perfect in cold weather and technical ice/snow. 聮s AFS 8000s ($440) would work well, too.
Scarpa Summit GTX Mountaineering Boot

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As for the trekking portion, well, I don聮t know. Plastic boots have come a long way in recent years, but they聮re still not the most comfortable thing to wear for long trail trips. They聮re heavy and rather stiff. But then again, you don聮t really want to have a pair of plastics in your pack to lug around.
So, you could buy a pair of plastics and do some hiking around New York in them to see how they feel. You might find you聮re OK with using them as a hiker. Alternatively, you could buy a pair of somewhat newer-age boots that use materials that offer more comfort than plastic but have insulation and vertical-ice capability. For example, you might like the Trango Extreme EVO Light ($400). It聮s an insulated, light, technical boot made with synthetic leather and other tech-y stuff to yield a boot that can handle trails, ice, rock, you name it.
聮s Summit GTX ($390) is another excellent all-arounder. Maybe even better on the trail than the Trangos, with leather uppers, a Gore-Tex liner, and insulation. They are probably what I would take, if you want to know. I don聮t think you聮ll be doing anything super-technical. But you want a good mountaineering boot that is warm and can handle crampons well. The Summit would do the trick.
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