I've never seen any of the forest roads plowed in winter in the Breadloaf area, and doubt they will be unless they're logging. As for Lincoln Gap - there's a barricade placed just a bit beyond the last house (before the truly steep road begins). The barricade won't prevent you from driving around it, but if you do you're really on your own. Hard to overstate the steepness of that road. People often do trek up the road to the top of Lincoln Gap, but it's almost always from the other side.
As for Jay - there are a couple of pull-outs in the area on either side of the road, not too far from where the LT crosses it. Have used both at one time or another for day hikes. As for overnights in winter - you might call the VT AOT garage in the Troy area and ask them if it's OK. My hunch is that it is, particularly if they know the vehicle isn't abandoned.
As I recall, there's a 3-sided leanto on the northboard side of the LT not too far from the highway. Might be on a short spur trail.
PS - Just saw Peak's post. You might also call the GMC in Waterbury. As I recall, Ben Rose, the director of the GMC, helped lay out the Catamount Trail not too long before he took the position at the GMC.