Best route - Maine to Cape Hatteras

That is a tough one.

Yes, give NYC as wide a berth as possible.

295 through NJ is surprisingly easy, depending on time of day.

The other way is to skirt around Philly to the west then come down the DELMARVA and VA Beach over the CBBT, an experience all by itself. Where else can you drive 60' under the ocean?

From the Tappan Zee either the NJ Turnpike to the Delaware Memorial Bridge or the Garden State Parkway to the Cape May - Lewes Ferry. Then right down the Delmarva as suggested above. Not perfect but just avoid likely heavy traffic hours

It pretty much depends on how you are planning on traveling. Are you doing it all in one day? Stopping somewhere halfway? Willing to drive a little farther to avoid sitting in traffic?
This is the route I would prefer. Long easy way around

If you have a trailer you should not plan a Garden State Parkway Route: Tractor trailers are permitted south of Exit 105, provided that they meet other restrictions (maximum 40-foot length for trailers and 55-foot length overall). Trucks and other commercial vehicles are prohibited from the Garden State Parkway from Exit 105 in Tinton Falls north to the New Jersey-New York border.

Going North on a weekend I have done the George Washington Bridge & the Tapen Zee Bridge both are a lot of traffic. Once you get over Tappen Zee is just as bad on 84. I just look for what seems lighter that day I'm going. You just should stay away from the parkway.

Going South if you have a trailer you can't take the chesapeake tunnel You have to take the inner loop of the Francis Scott Key Bridge to continue to go South.