I'm wondering what can I reasonably negotiate?
Everything is negotiable. Whether the company will give you what you are asking for depends on the company, the expense, the standard practices, your standing in the company, the likelihood of your leaving the company, etc.
Most companies where I have worked wouldn't give you any of the things you have listed other than permission to work from the New York Office. They may feel that they are granting you the perk of working from an office that you prefer. They may not feel that they should owe you anything else due to your desire to move. Asking for these things makes you more expensive, and may make it more likely that they would just deny your request to work remotely.
In general, you would have more leverage asking to negotiate when you are being imposed upon by the company, rather than when you are asking to move to be closer to your family.
Here you are basically saying "I would like to move to be closer to my family. Now please give me things (money, expenses, trips, etc)." Anything's possible, but you are less likely to receive these things than if the company was asking you to move.
Consider ahead of time what your response will be if the company says "No" to any or all of your requests, before you meet to negotiate.
I'm wondering if it's reasonable to ask for a cost of living
adjustment as well?
Companies often will give a cost of living adjustment when they ask the employee to relocate to a more expensive area so that the employee isn't financially disadvantaged by the move. An adjustment could be granted, but it's less common when the employee simply wishes to relocate based on their own initiative.
Ask, and see what kind of response you get. Consider the advantages to the company for your move, and why they should consider paying more.
What about moving expenses?
Again, if the company wanted you to move, then it would be reasonable to expect them to help pay for the move. But in this case, it's you who are asking to move.
Ask, and see what happens. Don't be surprised if the answer is "No".
What about travel back to Northern Virginia every other week (easy
train ride)?
Other than work-related travel that would be very unusual, in my experience. Travel back is typically granted when an employee is asked to separate from their family or extended family. Why should the company pay for your travel back to the area you chose to leave? You are asking to move so that you can be closer to your family.
You could ask, but this would be a surprising benefit if granted, in my opinion.
Work-related travel back to the Northern Virginia site would likely be reimbursed. Whether than means every other week or not depends on the company/department needs, I imagine. In places where I have worked, voluntarily remote workers hardly ever traveled back to the home office (perhaps once per year or so).
If it was necessary to travel back every 2 weeks, that may be an argument not to work remotely.
