The trail from Four Corners down to Panther Gorge, then up Haystack, is generally not travelled in winter. I would say the odds of your finding it broken are small.
As for the other segments of your planned route, some of these are also difficult to predict with accuracy, but what the heck, I'll give it a try anyways. Assuming no big snowfall for a week prior to your hike: the trail into Marcy will certainly be broken, as will the side trail to Phelps. Tabletop trail is somewhat less certain, although the odds lean in your favor since it is an "easier" mountain and a regular target for early-season ascents. Over Marcy to Gray and Skylight the odds start going slightly against you, especially since it is early season and these are more remote summits that are generally more favored for later winter ascents, and also the more favored route is from Lake Colden. The Phelps trail from Little Marcy down to the Range connector trail is occasionally traveled in winter but not regularly, as most winter Marcy ascents are done either from the Loj or Lake Colden. Even odds the connector trail up to Haystack will be broken. Haystack to Basin to Saddleback odds are somewhat against you, then Saddleback down the Range to LWJ your odds are good.
The itinerary you describe is ambitious. In summer travelling light, a well-trained hiker/runner (top 5%), familiar with the trails and all the watering holes, can cover this route, excluding Sawteeth, in perhaps 15 hours. I say this not to discourage you (those who know me will agree that I am among the last to discourage ambitious projects), but to provide a frame of reference from which you can extrapolate and fine tune your specific plan.