sealing off 1 entrance in a duplex?

This is a question you might ask the managing agent for the building and the building super who may have knowledge if others have done this. You also have to be sure the proposed change complies with fire regulations. As for the seller being "all over you" to sign, that should be put in the category of "irrelevant" concerns. This is perhaps the largest expenditure of your life. You commit to making it when you are ready to, not when someone else tells you to. NOTHING is more important than your due diligence on this purchase and that includes being as sure as you can that the unit is suitable for the purposes to which you intend to put it.

Getting real answers is going to take time and conversations, and involve several professionals. At this point it sounds like you may prefer to proceed anyway and accept the risk that you have to keep the layout as-is.

Here are the typical considerations:

Building codes and emergency egress: the hallway door upstairs is one of your exits for that level. Depending on the code details and the unit's internal stairs, you might not be allowed to remove that exit. A good architect could advise on the compliance aspects.

Hallway aesthetics: The coop may want the door maintained rather than letting you modify their hallway wall to remove it. If the egress concerns are satisfied, though, you should be able to build a false wall on your side of the door to hide it without removing it. It will cost you a few square feet, no big deal. Or make it the back 'wall' of a closet and lock it up tight.

Plumbing rules: this is a coop thing; look elsewhere in this forum for extensive discussion of wet over dry. The only mitigating factor in your case is that your new bathroom/laundry would be leaking into your own apartment below, rather than another tenant's, but it may still be rejected on those grounds. The coop's engineer would also review the impact of your additional fixtures on the riser piping and overall capacity.

A change to egress must comply with the building code. Whether you'll be able to close off the second floor entrance will largely depend on the internal staircase in the apartment (width, riser height, etc.). You should expect to be required to have plans drawn up by a registered architect and approved by the Dept of Buildings before you're allowed to modify the entrance.

Most duplex apartments that I've seen wouldn't be code compliant for single floor egress.

Right, it's a fire code issue. Both floors of a duplex have to have an exit to the hall. You might be able to block yours off as long as nobody inspects, but the co-op or condo won't go for it. There's too much liability for everybody if someone dies in a fire.

Look at http://streeteasy.com/nyc/sale/849343-coop-54-west-82nd-street-upper-west-side-new-york

That one has a third exit, up to the roof, but that upper-floor door isn't there just to get furniture in and out.

The only chance you would have to be able to seal off one of the two egresses legally is if your building has a window fire escape.

To the extent that the door has to be in a "dry" area already, it may well be that you're also limited in the ability to convert it into a true part of the bathroom in addition to the means of egress issues raised above.

That being said, I have seen some pretty impressive solutions in duplexes where the means of egress is hidden in a closet, behind a stack of shelves or through a "secret" door. This could be a potential solution.