I've done everything on the east shore except Buck mountain (just haven't gotten to it yet).
However, all of those hikes were in the spring/summer/fall. During the winter the access roads are likely to be maintained because of year-round residences, but how well is a good question.
Because this is a more 'touristy' spot, there's much less traffic during the off winter season. I can tell you that most of the Adirondacks, including the Saratoga and Lake George regions, have received very little snow this season. The trails are mixed hard packed to ice to no snow at all. You could end up using crampons, snowshoes, and bare boots all in the same mile.
There's plenty of lean-tos and places to pitch a tent. I enourage you to obtain a copy of the Eastern ADKs guide book from the Adirondack Mountain club. It includes a map and would answer most of your more specific questions. I don't recommend hiking this area without a map as there are a lot of trails and private trails that could get you lost, and if anything exhausted from back-tracking.
Black mountain is one of the more heavily travelled summits on the eastern shore, and because there hasn't been a ton of snow it won't matter if someone's already broken trail. The grade is moderate most of the way. The problem with the approach is that you're coming in from the east/northeast which will tend to have the most runoff and freeze, causing the most ice. I do remember there being a lot of water heading up that trail, so that's a good indicator to me that it will likely be icy. Heading down the other side towards the lake is very steep, but because the sun hits it most of the day it should be mostly bare-rock.
